Barely a week ago, Israelis rallied in Tel Aviv’s Hostage Square, waving placards thanking President Trump and his Middle East envoy for their role in helping secure an initial cease-fire deal in Gaza and getting some hostages freed.

Many of them were hoping that Mr. Trump would strong-arm Israel’s long-hesitant prime minister, Benjamin Netanyahu, into agreeing to negotiate an end to the war with Hamas and get the rest of the hostages released when the two leaders met in Washington on Tuesday.

Instead, they woke up to news of Mr. Trump’s fantastical idea of removing the population of roughly two million Gazans from the devastated enclave to make way for a glittering, American-owned Middle Eastern Riviera.

Far-fetched as Mr. Trump’s vision for Gaza may be — the Arab world has roundly rejected it and any forcible removal of a population violates international law — it abruptly shifted attention away from the future of the cease-fire deal, whose initial, six-week phase is due to end in early March.

As Mr. Trump sketched out his grandiose plans for Gaza, he placed little public pressure on Mr. Netanyahu to proceed with talks via Qatari and Egyptian mediators to turn the temporary cease-fire into a permanent cessation of hostilities. That left Israel with a wide berth on how it might deal with Gaza next.

The talks, which were meant to start this week, are now up in the air. And Mr. Netanyahu will leave Washington with Mr. Trump’s endorsement of what far-right members of the Israeli government have effectively been calling for: the mass migration of Palestinians from Gaza.

That leaves the fate of the hostages still held by Hamas in question as the militant group assesses how to move forward and many Palestinians worrying about whether the war might resume again.

“On the one hand, we are very grateful for what Trump has been doing,” said Idit Ohel, whose son, Alon Ohel, 23, was kidnapped from a bomb shelter as he tried to flee a music festival during the Hamas-led assault of Oct. 7, 2023, which started the war.

“Now,” Ms. Ohel said of Mr. Trump, “I don’t understand the implications of what he is saying or how this is going to bring my son home.”

Mr. Netanyahu, in an interview with Fox News late Wednesday, hailed Mr. Trump’s idea as “remarkable,” saying it should be “pursued,” drowning out any talk of the details of how to move cease-fire negotiations forward.

And on Thursday morning, Mr. Netanyahu’s loyal defense minister, Israel Katz, issued a statement saying he had instructed the Israeli military to prepare a plan to facilitate the exit of “any resident of Gaza who is interested to leave to any place in the world that agrees to accept them.”

The initial phase of the cease-fire deal took effect on Jan. 19 and provides for Hamas to release 33 hostages in exchange for hundreds of Palestinians imprisoned in Israel. About 79 hostages remain in Gaza, at least 35 of whom are presumed to be dead.

Talks were due to start on Monday — Day 16 of the deal — on a second phase, which is supposed to result in the rest of the living hostages being released and to usher in a permanent cessation of hostilities. That would mean a full withdrawal of Israeli troops from Gaza.

Wording about the transition to the second phase had been left intentionally vague, since Israel and Hamas are holding out for mutually exclusive demands.

Mr. Netanyahu has vowed to continue the war until Hamas no longer holds sway in Gaza and to resume fighting, if necessary. Hamas refuses to give up control or disarm.

In repeated statements in Washington, Mr. Netanyahu laid out his three priorities for Gaza, with the hostages only coming in second.

“In Gaza, Israel has three goals: Destroy Hamas’s military and governing capabilities, secure the release of all our hostages and ensure that Gaza never again poses a threat to Israel,” he said.

Mr. Netanyahu could stand to lose his own grip on power, with the far-right flank of his governing coalition having threatened to quit if he ends the war in Gaza with Hamas still in control there.

As of Thursday, no Israeli delegation had yet set out to Doha, Qatar, for negotiations, according to two Israeli officials who were not authorized to discuss the sensitive issue publicly.

Mr. Trump also sounded less committal than he has in the past about the fate of the hostages and ending the war, saying it was unclear if the cease-fire would hold. But he spoke of “going to a Phase II” of the cease-fire and said he would like to get all the hostages out. “If we don’t, it will just make us somewhat more violent,” he said, possibly indicating U.S. backing for a resumption of the fighting.

In the Middle East, analysts were parsing what Mr. Trump’s tectonic diversion on the future of Gaza might mean in the more immediate term.

“I think what he did was throw the old checkers board off the table and replaced it with Monopoly,” said Kobi Michael, an expert in the Israel-Palestinian conflict at the Institute for National Security Studies at Tel Aviv University. “He didn’t just change the rules of the game but the game itself,” he said.

Both Israel and Hamas are likely to want to buy time — Hamas to rehabilitate itself and its forces after 15 grueling months of war and Mr. Netanyahu to keep his right-wing coalition together — and may try to extend the first phase of the deal, allowing for more hostage-for-prisoner exchanges.

Mr. Michael said Mr. Trump’s vision for a Gaza without Gazans could work as a threat and put significant pressure on Hamas to release more hostages. Conversely, he said, it could cause Hamas to walk away from the deal altogether.

“Mr. Trump is a businessman,” Mr. Michael said. “He takes risks.”

Zakaria al-Qaq, a Palestinian expert in national security, said that even the mere suggestion of relocating two million Gazans was likely to complicate the cease-fire negotiations by making Hamas more cautious, and to destabilize the entire Arab world.

Mr. Trump’s declaration, he said, was “The perfect recipe for recruiting more people to Hamas,” adding that Mr. Trump’s “new colonialism” had given Hamas “easy marketing tools.”

Many people believe Mr. Trump’s vision for Gaza is not feasible, but regardless of the reality, Mr. Netanyahu has come out with no sign of being pressured by Mr. Trump, or of any daylight between them. His government is intact, for now.

An Israeli official who briefed Israeli political reporters in Washington after the Netanyahu-Trump meeting said it was now clear to Mr. Netanyahu’s coalition partners that bringing down his right-wing government with Mr. Trump as president would be irresponsible and foil “historic” opportunities in the coming years.

Relatives of the hostages warn that they do not have time.

“I live in daily fear,” said Alon Nimrodi, the father of Tamir Nimrodi, an Israeli soldier who is slated to be released only under a second phase of the deal.

Mr. Trump’s vision for Gaza was not a bad one, Mr. Nimrodi said. “But this is not the time to talk about it,” he said. Plans for Gaza should wait till “after the hostages are out,” he said.