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It is “an absolute all-out onslaught of pressure,” one of the people said.

Sharpen Haskel, one of several lawmakers who left Mr. Netanyahu’s Likud party last year to join the New Hope party, said Tuesday she had “many approaches from Likud members and supporters” to defect. But she said she didn’t intend to.

A Likud spokesman said “we are making every effort” to prevent the establishment of the rival coalition’s government.

The defection of just one or two lawmakers could prevent the new coalition from mustering a majority and force another election—what would be the country’s fifth since 2019.

President Biden and Israeli Prime Minister Benjamin Netanyahu have been working together for decades, but their friendship faces new trials amid the Israel-Hamas truce. WSJ’s Gerald F. Seib takes a look at key moments from the last 15 years of their relationship, and what might come next. Photo illustration: Todd Johnson

Naftali Bennett’s Yamina party is expected to be the main partner to Yair Lapid’s centrist Yesh Atid party in the coalition. Besides the right-wing parties, others that are likely to be part of a new government are left-wing Labor and Meretz and centrist Blue and White, as well as Israel’s Islamist party Raam.

Mr. Lapid received the mandate to form a government in early May after Mr. Netanyahu first failed to do so following an inconclusive March election, the country’s fourth since 2019.

The politicians are racing to agree on the government’s agenda and cabinet portfolios ahead of a midnight Wednesday deadline. Aside from their common goal of ousting Mr. Netanyahu from power, the parties say they can find common ground on issues such as the economy, infrastructure and fighting crime.

Mr. Netanyahu enjoys support among Israel’s right-wing and religious voters, which has helped him maintain a grip on power since 2009. Soon after Mr. Bennett on Sunday said he planned to join Mr. Lapid in a coalition, Mr. Netanyahu in a televised statement said such a government would be a danger to the security of Israel and to the future of the country.

“I call on lawmakers voted in by the right-wing—do not deceive your voters or yourselves,” Mr. Netanyahu said.

Overnight Sunday, Mr. Netanyahu’s Likud placed three large banners on a building at the entrance to Jerusalem, warning his right-wing opponents that “the right will not forgive you.” Mr. Netanyahu’s supporters have also organized protests outside the homes of right-wing lawmakers.

Israel’s police have put Mr. Bennett and another of his party’s leaders under extra security due to increasing threats of violence against them.

Mr. Netanyahu’s rivals say he is focused more on staying in office to help shield himself from the allegations of corruption than giving priority to the needs of the country. He denies any wrongdoing.

Mr. Bennett, a former defense and education minister, said he has no choice but to align with the new coalition and halt the destructive cycle of elections. “The option of a right-wing government headed by Netanyahu simply doesn’t exist,” he said.

Under the emerging terms of the coalition deal, Mr. Bennett, a longtime standard-bearer for Israel’s right and former aide to Mr. Netanyahu, would serve first as prime minister for two years and then Mr. Lapid would succeed him.

Mr. Bennett controls seven seats in the Knesset, or parliament, to Mr. Lapid’s 17, but right-wing Jewish Israelis who previously supported Mr. Netanyahu’s governments may find him a more acceptable candidate as prime minister.

Mr. Netanyahu and his party have accused the right-wing leaders, especially Mr. Bennett, of stealing right-wing votes to create a government without him. “What is happening here is against the will of the people,” Miki Zohar, a Likud member, told Army Radio on Tuesday.

Mr. Zohar also said it was “bizarre and totally absurd” that Mr. Bennett, whose party won just seven of 120 seats in parliament, would become prime minister.

Mr. Netanyahu’s Likud party, which won 30 seats, publicized an offer made to both Mr. Bennett and New Hope leader Gideon Saar, whose party has just six seats, to serve as prime ministers with Mr. Netanyahu in a triple rotation government.

On Tuesday morning, the Likud party publicized letters it sent to the legal advisers of Israel’s president and parliament, arguing that Mr. Lapid must serve first as prime minister because he has the mandate to form the government and can’t hand over the role to Mr. Bennett. The legal adviser to the presidency rejected the claim.

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