For months, Prime Minister Benjamin Netanyahu of Israel has refused to offer a timeline for ending the war against Hamas in Gaza, a stance his critics see as a political tactic. But he has been put on the spot by President Biden’s announcement outlining a proposal for a truce.

Mr. Netanyahu, a conservative, has long juggled competing personal, political and national interests. He now appears to be facing a stark choice between the survival of his hard-line, hawkish government and bringing home hostages held in Gaza while setting himself and Israel on a new course away from growing international isolation.

Critics of the prime minister have portrayed him as indecisive and say there are two Netanyahus. One, they say, functions pragmatically in the small war cabinet he formed with some centrist rivals, to give it public legitimacy. The other is effectively being held hostage by the far-right members of his governing coalition, who oppose any concession to Hamas and who ensure his political survival.

On Friday, Mr. Biden outlined broad terms that he said were presented by Israel to the American, Qatari and Egyptian mediators who have been pushing for a deal to pause the fighting and free hostages in Gaza. Israeli officials confirmed that the terms matched a cease-fire proposal that had been approved by Israel’s war cabinet but not yet presented to the Israeli public.

Now, analysts say, it is crunchtime for Bibi, as the prime minister is popularly known.

Mr. Biden “booted Netanyahu out of the closet of ambiguity and presented Netanyahu’s proposal himself,” Ben Caspit, a biographer and longtime critic of the prime minister, wrote in Sunday’s Maariv, a Hebrew daily. “Then he asked a simple question: Does Bibi support Netanyahu’s proposal? Yes or no. No nonsense and hot air.”

The leaders of two far-right parties in the coalition — Bezalel Smotrich, Israel’s minister of finance, and Itamar Ben-Gvir, the national security minister — have pledged to bring Mr. Netanyahu’s government down if the prime minister goes along with the deal outlined by Mr. Biden before Hamas is fully destroyed. Some hard-line members of Mr. Netanyahu’s own Likud party have said they will join them.

At the same time, Benny Gantz and Gadi Eisenkot, two former military chiefs who joined the emergency government for the duration of the war, have threatened to withdraw the support of their centrist National Unity party by June 8 if Mr. Netanyahu fails to come up with a clear path forward. And opposition parties have begun organizing to try to topple the government.

The cease-fire proposal involves three phases. Under the plan, groups of hostages would be released in exchange for hundreds of Palestinian prisoners in Israeli jails, a temporary cease-fire would turn into a permanent cessation of hostilities, and an internationally backed effort would be launched to rebuild Gaza.

More than 100 hostages were released under a more limited deal last November. An estimated 125 people are still being held by Hamas and other armed groups in Gaza, though dozens are believed to be dead.

Israelis were left to parse two statements following Mr. Biden’s speech that Mr. Netanyahu’s office put out — unusually — during the Sabbath. The statements neither forcefully endorsed the proposal nor denied that it had been presented by Israel to the mediators. Conditional and open to interpretation, they were seemingly designed to leave Mr. Netanyahu’s options open.

The first statement said that Mr. Netanyahu had authorized Israel’s negotiating team to present a proposal that would bring about the release of the hostages and also “enable Israel to continue the war until all its objectives are achieved, including the destruction of Hamas’s military and governing capabilities.”

The second reiterated those conditions for ending the war and added: “The notion that Israel will agree to a permanent cease-fire before these conditions are fulfilled is a non-starter.”

Notably absent, though, was Mr. Netanyahu’s oft-stated goal of “total victory” over Hamas in Gaza — a slogan that Mr. Biden on Friday dismissed as a vague objective that would mean indefinite war.

While some of the staunchest supporters of Mr. Netanyahu have taken to wearing blue baseball caps emblazoned with a “Total Victory” logo, the statements from the prime minister’s office over the weekend appeared calibrated to fall in line with the war goals articulated by the Israeli military and defense establishment, as well as by Mr. Biden.

On Sunday, Defense Minister Yoav Gallant, reaffirmed that “in any process to bring about the end of this war, we will not accept the rule of Hamas.” He said that Israel would “isolate areas” in Gaza, clear them of Hamas operatives and “introduce forces that will enable an alternative government to form,” without elaborating on who those forces might be.

Mr. Netanyahu’s opponents have accused him of prolonging the war to stave off an election and a public reckoning for the Israeli intelligence and policy failures leading up to Hamas’s devastating Oct. 7 assault on Israel. That attack set off Israel’s military offensive in Gaza, and the widespread death and destruction that followed.

But much now hangs in the balance for Mr. Netanyahu, who finds himself at a crossroads both political and strategic.

Since forming his current government — the most right-wing and religiously conservative in Israel’s history — 17 months ago, Mr. Netanyahu has had an increasingly testy relationship with Mr. Biden. And while the four top congressional leaders formally invited him to address a joint meeting of Congress on Friday, they did not set a date.

The bipartisan unity masked a fraught behind-the-scenes debate over receiving the prime minister, given the deep political divides in the United States over Israel’s prosecution of the war in Gaza.

Mr. Biden presented the truce deal as not just a way to stop the bloodshed in Gaza but also as a pathway to a grander Middle East bargain that could lead to Israel becoming more integrated into the region and include a “potential historic normalization agreement with Saudi Arabia.” Israel, Mr. Biden said, “could be part of a regional security network to counter the threat posed by Iran.”

Mr. Biden acknowledged that some parts of Mr. Netanyahu’s coalition would not agree with the proposal and would rather keep fighting for years and occupy Gaza. He exhorted Israeli leaders “to stand behind this deal, despite whatever pressure comes.”

Israel’s president, Isaac Herzog, said on Sunday that he would give Mr. Netanyahu and the government his full support for a deal that would bring the hostages home. Though the president’s role is mostly ceremonial and he lacks executive powers that might help Mr. Netanyahu should his government fall, his voice is supposed to be a unifying one that reflects the national consensus.

After Mr. Netanyahu’s office issued the statements about the truce proposal, John Kirby, a U.S. National Security Council spokesman, sought to dispel any ambiguity about is origins. “This was an Israeli proposal,” he said on ABC News on Sunday. “We have every expectation that if Hamas agrees to the proposal, as was transmitted to them — an Israeli proposal — then Israel would say yes.”

Still, drawing on past experience, some Israeli analysts remained skeptical that Mr. Netanyahu would be willing to ditch his hard-right coalition partners. That, said Reuven Hazan, a professor of political science at the Hebrew University of Jerusalem, would require “a new Netanyahu.”

“Whenever he’s had the choice of going for what’s good for the country or for his extremist fanatics, or even for his own party, he’s always gone with his extremist fanatics,” Professor Hazan said. He said that Mr. Netanyahu had also learned how to say “Yes, but …” to the Americans, then “wait for Hamas to say ‘No’ and drag it out as long as possible.”

For its part, Hamas said in a statement on Friday that it viewed Mr. Biden’s speech “positively,” and expressed readiness to deal “in a constructive manner” with any proposal based on a permanent cease-fire and other terms he outlined.

Given the American political timetable, Professor Hazan said, Mr. Netanyahu needs only to engage in “survival politics” until Labor Day at the end of the summer, after which the administration will be focused on the November presidential election.

“Is Netanyahu ready to make a 180-degree turn and do what is good for the country?” Professor Hazan said. “Everyone’s in a tizzy about this now. Don’t hold your breath,” he cautioned. “A speech by President Biden doesn’t mean we have a new Netanyahu.”