TEL AVIV—Since taking over in June as Israel’s prime minister, Naftali Bennett has tried to construct a new political order. The biggest obstacle might be the man he displaced, his predecessor, Benjamin Netanyahu.

Mr. Bennett, a right-wing politician who will be in Washington this week to meet President Joe Biden, heads an unstable coalition of leftist, right-wing and Arab parties and by necessity must govern from the center.

He has tried to call every minister each week to ask how they are, according to coalition members. His tweets add the handles of other ministers to congratulate them on the government’s work, which is studiously avoiding tackling contentious issues head on. He’s turned to private sector experts on the coronavirus and other issues, asking for ideas via WhatsApp.

In a video published on Twitter this month, Mr. Bennett and members of each coalition party urged people to get vaccinated against the coronavirus, regardless of political affiliation.

Mr. Bennett’s coalition accomplished what no other opponent of Mr. Netanyahu could over the past 12 years—it drove him from power. The political survival of the 49-year-old prime minister depends on winning enough votes beyond his traditional constituency to continue to lead. At stake is moving the country past a divisive and turbulent political period, with four elections in two years

Mr. Netanyahu, 71 years old, has set out to topple the coalition and accuses Mr. Bennett of stealing conservative votes to form a “feeble and submissive government.”

Benjamin Netanyahu, Israel’s former prime minister, has been critical of Naftali Bennett’s handling of coronavirus.

Photo: EMMANUEL DUNAND/Agence France-Presse/GETTY IMAGES

He argues the new government’s policy of open communication with the U.S. inhibits Israel’s ability to act independently, in particular, to prevent Iran building a nuclear weapon. In Israel’s parliament, known as the Knesset, Mr. Netanyahu has united most of the opposition parties to vote against any bill put up by the government, no matter its content.

Because the coalition controls just 61 of 120 seats, Mr. Netanyahu’s strategy means many votes require a scramble to ensure not a single lawmaker breaks ranks. The coalition’s biggest test will come ahead of a Nov. 4 deadline to pass a budget, a vote that if the government loses, triggers new elections.

“We can make this government fall very, very soon,” said Miki Zohar, a member of the Likud party and a Netanyahu ally.

Mr. Bennett and Mr. Netanyahu declined interview requests.

Mr. Netanyahu recently helped to convince a member of Mr. Bennett’s Yamina, Amichai Chikli, to vote against his party. Despite an all-night lobbying effort, the prime minister recently failed to bring his parties together for a crucial vote to renew a law that effectively barred Palestinians married to Israelis from gaining citizenship. And as coronavirus cases rose in recent weeks, Mr. Netanyahu criticized Mr. Bennett for upending the country’s aggressive vaccination campaign.

Mr. Netanyahu, and Likud member Miki Zohar. Mr. Zohar has said, “We can make this government fall very, very soon.”

Photo: ronen zvulun/Reuters

For 12 years, Mr. Netanyahu was the dominant personality in Israel’s government with support from right-wing and ultraorthodox parties. At times, he held the foreign, defense, communication and other ministries, frustrating allies who felt sidelined. Some started their own parties and now sit in Mr. Bennett’s coalition.

“We’ve got eight different parties, left and right, Jews and Arabs,” Mr. Bennett said in a speech July 6 at the U.S. Embassy in Jerusalem. “It’s working because there’s a spirit of goodwill.”

To bind the coalition together, Mr. Bennett is trying to first address issues on which the partners can agree, such as tackling the coronavirus and Iran, and plans to leave sensitive topics like the Israeli-Palestinian conflict for later.

He’s received support from the Biden administration, which has held off reopening the U.S. consulate for Palestinians in Jerusalem to avoid causing strife in Mr. Bennett’s coalition, according to a senior Palestinian official and another person familiar with the matter. A spokeswoman for the U.S. Embassy in Jerusalem declined to comment on the delay, and pointed to comments made by U.S. Secretary of State Antony Blinken in May that the consulate would reopen, but that no date had been set yet.

Mr. Bennett is scheduled to meet with Mr. Biden this Thursday at the White House. The Israeli leader said he wants to focus on Iran’s nuclear program. A White House announcement made clear that peace efforts between Israelis and Palestinians would be discussed. The statement from Mr. Bennett’s office made no direct mention of the issue.

The prime minister’s conservative religious Yamina party is one of eight in the coalition. It includes Yesh Atid, a centrist faction that promotes socioeconomic policies, and Blue and White, formed two years ago in opposition to Mr. Netanyahu. The coalition’s New Hope party supports Israeli settlement in the occupied West Bank, alongside Yisrael Beiteinu, a nationalist party, which promotes greater separation of religion and state.

Meretz, a far-left faction, and center-left Labor, once Israel’s dominant player, both support the establishment of a Palestinian state. Ra’am is the first independent Arab party to declare support for a government and seeks to help the country’s disadvantaged Arab minority.

With these eight parties all holding enough seats to topple the new coalition, Mr. Bennett can’t govern as presidentially as Mr. Netanyahu, said Yohanan Plesner, president of the Israel Democracy Institute, a Jerusalem-based think tank, and former parliamentarian. “Bennett by design needs to collaborate,” Mr. Plesner said.

In a recent poll of roughly 1,000 Israelis published in newspaper Maariv and conducted by the Pnima Movement, a nonprofit that seeks to narrow Israel’s political divides, a fifth of respondents said they hated leftists, including 30% of those who identified as right-wing. A fifth of people who identified as left-wing said they hated settlers in the occupied West Bank; a quarter of respondents said they hated the ultraorthodox.

At the heart of Israel’s new government is a partnership between Mr. Bennett and Yair Lapid, the foreign minister whose Yesh Atid party won the second-most seats in March’s election. Mr. Lapid offered Mr. Bennett the prime minister portfolio in a rotation agreement under which Mr. Lapid will become leader in two years, if the government survives.

Israeli Prime Minister Naftali Bennett and Foreign Minister Yair Lapid. Mr. Bennett has reached out to coalition members as he attempts to build consensus.

Photo: gil cohen-magen/Agence France-Presse/Getty Images

At the opening in June of the Honduras Embassy in Jerusalem, Mr. Bennett introduced Mr. Lapid to the country’s visiting president on the sideline of the event and added, “He’s foreign minister but he’s also my brother,” according to people familiar with the exchange.

A poll published Friday by Israel’s Channel 12 News found that in race between Mr. Netanyahu and Mr. Bennett, 45% believed Mr. Netanyahu was most suitable to be prime minister compared with 25% for Mr. Bennett, with the remainder preferring neither. The same poll found 40% of Israelis believe Mr. Bennett is doing a good job while 51% said he is doing a bad job. And among right-wing voters, 71% believed the era of Mr. Netanyahu’s leadership hasn’t yet ended.

A majority of Arab citizens of Israel, who make up around 20% of the population, graded Mr. Bennett’s performance so far as poor or average, according to polling at the end of July by the Israel Democracy Institute.

Still, Mr. Bennett’s coalition partners have bought into his consensus leadership to keep Mr. Netanyahu on the sidelines. Esawi Frej, an Arab minister from the Meretz party who manages the government’s relationship with the Palestinians, said he was overcome with emotion when the new prime minister in an early cabinet meeting opened his remarks by declaring the fight against violent crime in Arab towns a national mission.

“It’s a new way,” Michal Rozin, the parliamentary chair for Meretz, said. “The right will understand we are not evil and don’t want to help our enemies to destroy Israel, and our people understand that they are not all evil fascists who want to kill all the Arabs.”

While Mr. Bennett has said the government will avoid sensitive issues for now, the prime minister in recent weeks has begun to formulate a policy related to the Palestinians in the West Bank and Gaza following a school of thought promoted by Micah Goodman, a Jerusalem-based philosopher. Mr. Goodman is the author of a book that discusses “shrinking” the conflict by taking economic and security actions that boost the lives of Palestinians and avoids either side making concessions toward a long-term peace deal.

Using that framing, Mr. Bennett hopes to build goodwill between the two sides, and at the same time, unite his ideologically disparate coalition, according to Mr. Goodman, who has discussed the idea with the prime minister. “The formula of this government is that it can perform actions that they agree on without agreeing on the endgame,” Mr. Goodman said.

So far, the coronavirus has proven Mr. Bennett’s biggest challenge, sparking Mr. Netanyahu’s most vociferous criticism.

Israel now has one of the highest global per capita infection rates due to an outbreak of the Delta variant, and may need a fourth lockdown. This is particularly embarrassing for Mr. Bennett, who in a book he wrote last year titled “How to Defeat a Pandemic,” attacked Mr. Netanyahu’s policy of closures as a failure.

Mr. Bennett now says Israel’s Covid-19 booster shot campaign can prevent a lockdown. Mr. Netanyahu, on the other hand, says Mr. Bennett waited too long to get the campaign going, costing lives.

“How did you manage in such a short time to spoil our success?” Mr. Netanyahu asked the new prime minister in parliament last month.

An Israeli woman receives a third Pfizer-BioNTech Covid-19 vaccine. The country now has one of the highest global per capita infection rates due to an outbreak of the Delta variant.

Photo: Sebastian Scheiner/Associated Press

Israel’s severe outbreak has occurred despite around 80% of Israeli adults having had two shots of Pfizer’s vaccine. Health experts blame both waning vaccine protection and the highly contagious nature of the Delta variant. More than half of Israelis aged 60 and older have gotten their third shot since they became eligible this month. The country of 9.3 million has currently 692 cases of severe illness from the virus.

“We don’t need to call a press conference for every phone call with Bourla,” Mr. Bennett said in reference to Pfizer Inc. CEO Albert Bourla.

A week later, Mr. Netanyahu issued a statement saying he’d held phone calls with Mr. Bourla and Moderna Inc. CEO Stéphane Bancel to discuss vaccine supply to Israel that would allow the country to begin giving out booster shots—an outreach that political analysts said was designed to demonstrate his pull even in the opposition. Pfizer and Moderna declined to comment.

The acrimony between the two men belies a long relationship. Like Mr. Netanyahu, Mr. Bennett is a former member of Israel’s elite Sayeret Matkal special forces unit. He named his son, Yoni, after Mr. Netanyahu’s brother, who was killed during a 1976 hostage-rescue mission. When Mr. Bennett entered Israeli politics, after co-founding and selling an antifraud software company, he chose to serve as Mr. Netanyahu’s chief of staff in 2005.

He later entered the Knesset as head of the pro-settlement Jewish Home party in 2013 and began to challenge Mr. Netanyahu for influence over Israel’s right-wing voters. He became best-known for his opposition to the two-state solution, but struggled to challenge his mentor. Two years and three elections ago, Mr. Bennett’s party didn’t make it into parliament. At March’s election, he won seven out of 120 seats, compared with Mr. Netanyahu’s 30 members in Likud.

But Mr. Netanyahu couldn’t cobble together enough seats for a government. Some parties refused to sit in a coalition with him because he is on trial in an Israeli court for bribery, fraud and breach of trust, charges he denies.

Mr. Bennett, center, chairs the first weekly cabinet meeting of the new government in Jerusalem in June.

Photo: Emmanuel Dunand/Associated Press

Mr. Chikli, of Mr. Bennett’s Yamina party, was skeptical at the time about ideologically disparate parties working together, he recalled. He told Mr. Bennett each party would have the ability to collapse the government and hold it to ransom. Mr. Bennett replied that he should keep an open mind, Mr. Chikli said. On June 13, Mr. Chikli voted against the formation of the new government.

A few days later, Mr. Netanyahu invited the lawmaker to the official prime minister’s residence, which the former leader had yet to vacate. Mr. Netanyahu indicated that Mr. Chikli could join his party in a future election rather than run with Mr. Bennett, Mr. Chikli said.

Mr. Chikli’s vote proved crucial a few weeks later on July 6 when Mr. Bennett lost a test of his strategy when he tried to renew a law that has been extended every year since 2003 that effectively barred Palestinians married to Israelis from gaining citizenship. This time Mr. Netanyahu in an effort to embarrass Mr. Bennett, ordered his party to vote against, and the law was set to expire.

Just after midnight, as lawmakers debated the renewal, Mr. Bennett requested a meeting with members of left-wing Meretz, who were refusing to vote to renew the law.

Speaking in that session, Mr. Bennett said he believed the lawmakers love Israel, love their country and want the best for their country, according to Ms. Rozin. After agreeing to a compromise that allowed some Palestinians to gain citizenship, Meretz decided to back his government in the subsequent vote.

But the coalition lost the vote in a 59-59 lawmaker split. Mr. Chikli at the last moment joined Mr. Netanyahu’s Likud and voted against the government. Mr. Chikli says there is no quid pro quo between him and Mr. Netanyahu.

“It’s a war of attrition. Netanyahu is trying to fatigue different parts of the coalition,” said Abraham Diskin, a senior fellow with the Jerusalem-based think tank Kohelet Policy Forum. “Surviving is a very strong incentive that serves as a glue to this government.”

He added: “The other glue, of course, is the hatred of Netanyahu.”

—Fatima AbdulKarim in Ramallah contributed to this article.

Write to Rory Jones at rory.jones@wsj.com