As the war in Gaza grinds on, there is increasing talk of some “day after” formula for the broken territory. But that notion is an ephemeral one — there is not going to be a bright line between war and peace in Gaza, even if some sort of negotiated settlement is reached.

Israel has made it clear that it will not subcontract security along its southern border to anyone else, and Israeli military officials say their forces will come in and out of Gaza based on intelligence for a very long time to come, even after troops finally withdraw.

“The whole conceit of ‘the day after’ has to be retired,” said Aaron David Miller, a former U.S. official at the Carnegie Endowment. “It’s misleading and dangerous,” he said, because there will be no clear dividing line “between the end of Israeli military operations and a relative stability that allows people to focus on reconstruction.”

There are a variety of sketchy ideas — “plans” would be too specific a word — for what happens in the aftermath of hostilities. But there is a growing understanding that any sustainable settlement would require a regional deal involving countries like Egypt, Saudi Arabia, the United Arab Emirates, Jordan and Qatar.

Inevitably such a deal would have to be led by the United States, Israel’s most trusted ally. Most officials and analysts assume it would require new governments both in Israel and the Palestinian Authority, which partially governs the West Bank but is considered stale and corrupt, an indication of the long road ahead.

As a starting point, the American special envoy, Brett McGurk, is touring the region, his focus on “the potential for another hostage deal, which would require a humanitarian pause of some length to get that done,” according to a White House spokesman, John Kirby. Mr. McGurk will be joined in the coming days by C.I.A. Director William J. Burns, officials familiar with the talks said.

Mr. McGurk’s efforts are complicated, working through Qatar, which sends messages to Hamas leaders. Even with an agreement in principle between Israel and Hamas, the two sides will have to negotiate a phased exchange of hostages, women and children first, for Palestinian prisoners in Israeli jails.

To get all the hostages released, including soldiers, would require the controversial release of thousands of Palestinian prisoners, including those who have been convicted of murdering Israelis. Yahya Sinwar, the leader of Hamas in Gaza, was just such a case, let out in a previous prisoner exchange in 2011 after 23 years in jail.

Then there is the question of Mr. Sinwar and other Hamas leaders, if they are alive — will they go into exile as part of any settlement? For now, Hamas has rejected the idea.

But a first hostage deal “is the sine qua non of the administration’s larger regional deal,” said Martin S. Indyk, a former U.S. ambassador to Israel.

That, American officials hope, could open the way for broader negotiations. They would include moderate Sunni Arab states who have no great love for Hamas and its main backer, Shiite Iran, and who are concerned by Iran’s growing power.

While Prime Minister Benjamin Netanyahu of Israel supports efforts for a hostage deal, he is also campaigning for his political survival and has opposed a significant pillar of President Biden’s larger concept.

Mr. Biden has said that he would like a “revitalized Palestinian Authority” eventually running Gaza as a stage toward an eventual “two-state solution” — an independent Palestine, largely demilitarized, alongside Israel and committed to a lasting peace.

Mr. Netanyahu is portraying himself as the one person who can prevent the Americans from imposing a Palestinian state on a traumatized Israel or significant restrictions on the Israeli settlement activity in the West Bank that is gradually absorbing Palestinian land.

But the Americans believe they may have important leverage over Israel and Mr. Netanyahu to move ahead. Saudi Arabia, the key regional actor, has indicated that it wants to continue a path toward normalization with Israel in return for American security guarantees against Iran, itself a controversial demand.

But Saudi Arabia has also said that normalization, let alone any cooperation on a post-Gaza future, both in reconstruction and security aid, depends on the creation of an “irrevocable” pathway toward a Palestinian state, which Mr. Netanyahu rejects.

Mr. Netanyahu’s vision of a future Gaza is unclear. He continues to insist that Hamas will be “destroyed” and all the hostages released. But those goals seem more contradictory as the Israeli military operation in Gaza moves slowly and casualties on both sides mount, creating more domestic and international pressure on him.

He has stated what he does not want: Hamas to survive militarily and politically in Gaza; the Palestinian Authority to be given control over Gaza; any foreign peacekeepers; and an independent Palestinian state. He has denied wanting to reoccupy Gaza for the long run, but insisted that Israel retain security control over not just Gaza, but the West Bank as well.

Others have staked out positions on either side of Mr. Netanyahu.

His far-right partners, Bezalel Smotrich and Itamar Ben Gvir, have suggested displacing Palestinian citizens and resettling Gaza with Israelis. The idea is considered a non-starter and drew a specific American rebuke.

Opposition members of the current security cabinet, like Benny Gantz and Gadi Eisenkot, who are seen as popular alternatives to Mr. Netanyahu, are more likely to go along with the American idea of a larger regional deal, Mr. Indyk said.

So is the defense minister, Yoav Gallant, who has distanced himself from Mr. Netanyahu. All recognize that American support is indispensable for Israel, Mr. Indyk said.

Mr. Gallant, who is from Mr. Netanyahu’s Likud party, has laid out his own vague plan. He wants Israel to maintain security control of Gaza, with the military free to come and go as needed. He proposes that Egypt and Israel control Gaza’s southern border crossing together.

There would be no Israeli civilian presence in Gaza, in his vision, with civil administration run by Palestinians with foreign oversight, but not by the Palestinian Authority.

Mr. Gallant’s plan is thought to be similar to what Mr. Netanyahu privately thinks. But Mr. Gallant is also partly reflecting the Israeli military’s view, said Nahum Barnea, a well-connected columnist with the popular daily Yediot Ahronoth.

“The vision is not victory,” he said, but a managed intermittent conflict without a large permanent Israeli presence.

The military would like to turn Gaza into something akin to the situation in the restive, volatile northern West Bank cities like Nablus and Jenin, where it goes where it wants. In Gaza, it envisions operating from a buffer zone inside Gaza, now being constructed, and going deeper into the territory from time to time on specific operations.

The military, Mr. Barnea said, “is not looking for Somalia, but Nablus.”

No one thinks there is a quick deal to be done. To train some 6,000 Palestinian security forces to police Gaza, even in cooperation with some multinational Arab force, would take up to 10 months, American officials estimate.

In the meantime, they hope Arab countries, and possibly Turkey, heir to the Ottoman rulers of Gaza, would agree to police Gaza. That is a highly questionable aspiration, given the political sensitivity of Muslim nations policing Palestinians partly on behalf of Israeli security.

There is, then, no rapid path to an “R.P.A.,” the latest Biden administration acronym for a “revitalized Palestinian Authority.” At a minimum it would require the retirement or “emeritus” status of the Palestinian president, Mahmoud Abbas, internal reform and some form of Palestinian elections, senior American officials say.

The last elections were held in 2006, and new ones would almost surely result in some political role for Hamas. And there would have to be a temporary administration in Gaza made up of Palestinian notables or technocrats in the meantime, they say.

The Palestinians themselves are not ready. “To put it starkly, there is a complete disconnection between the international community’s call for a two-state solution and the willingness of Israelis and Palestinians to contemplate it now as a viable way to end their conflict,” Mr. Indyk said.

Still, he said, Washington “must try to fashion a new, more stable order in Gaza, and that cannot be done without also establishing a credible political horizon that leads eventually to a two-state solution.”

Despite the huge task for American diplomacy, time is limited — probably only until September, officials say — and that may create pressure to act. Mr. Netanyahu is conscious that Mr. Biden is up for re-election in November and may want to see what happens in the U.S. vote.

The Arab interlocutors are also acutely conscious that unless some sort of deal is done by autumn, they could be dealing with a lame duck Mr. Biden and awaiting the unpredictable Donald J. Trump. Even senior American officials think that the best chance for a deal is if Mr. Biden is re-elected, a senior Western diplomat conceded.

Yaakov Amidror, a former general and national security adviser, said he sees 2024 as a year of low intensity warfare. The next year or 18 months will be dedicated to finding and destroying Hamas tunnels, infrastructure and fighters, said Mr. Amidror, now a fellow at the Jerusalem Institute for Strategic Studies, a conservative think tank.

At the end, by mid-2025, he said, he believes Hamas will no longer have military and political capacity to run Gaza. And the Israeli army may be in a position to operate in Gaza along the lines of its West Bank model, he said.

So even with good intentions, there is a long road ahead to a true “day after,” and many possible ways for the best plans to fail. Prime among them may be, despite all American efforts, if war breaks out between Israel and Hezbollah in southern Lebanon, which could make the destruction in Gaza seem simply a prologue.

Reporting was contributed by Patrick Kingsley, Gal Koplewitz and Aaron Boxerman in Jerusalem and Vivian Nereim in Riyadh, Saudi Arabia.