As Secretary of State Antony J. Blinken sat down with Prime Minister Benjamin Netanyahu of Israel this week, American and Arab officials were expressing cautious optimism over the latest proposal from Hamas for a cease-fire in the Gaza Strip.

But just hours after speaking to Mr. Blinken, Mr. Netanyahu appeared more intent on delivering a fiery message aimed at his domestic audience. Instead of appearing side by side at a news conference with the secretary of state after they met on Wednesday — as is customary on such trips — the Israeli leader pre-empted him. Meeting on his own with reporters, he denounced the very proposal the Americans saw as a potential opening to a negotiated solution.

“Surrender to the ludicrous demands of Hamas — which we’ve just heard — won’t lead to the liberation of the hostages, and it will only invite another massacre,” Mr. Netanyahu said. Shortly after that, Mr. Blinken delivered his own, much more measured, assessment of the Hamas offer at a news conference in Jerusalem, saying that while it had “clear nonstarters,” it also left space for an agreement to be reached.

On Thursday, as Mr. Blinken ended his fifth visit to the Middle East in the four months since the war in Gaza began, it was clear that relations between the Biden administration and Mr. Netanyahu have become increasingly fraught. That raised questions about how drawn out the process might be to reach an agreement to end the conflict.

Mr. Blinken has been trying to secure a cease-fire in Gaza, a release of Israeli hostages held by Hamas and, eventually, a broader peace process for the region. But one stumbling block during his visit seemed to be the considerable domestic political pressures facing the Israeli prime minister.

Mr. Blinken tried to convey to Mr. Netanyahu that the United States, working with its Arab allies, was offering significant incentives for a peace deal. These include an openness to helping rebuild Gaza, as well as the prospect of formal diplomatic relations between Israel and Saudi Arabia, a regional powerhouse, if the Israelis agree to a process that leads to a Palestinian state and includes Palestinian governance of Gaza.

But if Mr. Netanyahu prioritizes his domestic audience in the negotiations with Hamas, he could test the patience of Arab leaders who are increasingly frustrated by the soaring number of Palestinian civilian deaths in Gaza.

“It will be up to Israelis to decide what they want to do, when they want to do it, how they want to do it,” Mr. Blinken said. “No one’s going to make those decisions for them. All that we can do is to show what the possibilities are, what the options are, what the future could be, and compare it to the alternative. And the alternative right now looks like an endless cycle of violence and destruction and despair.”

Despite the potential rewards of a peace deal, Mr. Netanyahu sounded intent on pressing on with the war.

The American-Israeli tensions resurfaced after Hamas, the armed Palestinian group that long controlled Gaza and led the Oct. 7 attack on Israel, responded to a plan to end the hostilities that had been worked out by the United States, Israel, Egypt and Qatar.

Hamas called for Israel to withdraw from Gaza, end its longstanding blockade of the territory, and free Palestinian prisoners in Israeli jails in exchange for the release of more than 100 Israeli hostages in Gaza. Israel holds more than 9,000 Palestinian prisoners, according to HaMoked, an Israeli rights group, citing data provided by the Israeli prison service.

Just hours after he met with Mr. Blinken, Mr. Netanyahu dismissed Hamas’s terms because Israel does not want to entirely withdraw from Gaza or allow Hamas to retain any control over the territory.

Mr. Netanyahu is also weighing the response of the Israeli public, large parts of which see Hamas’s defeat as a higher priority right now than a hostage deal. He must also assuage the far-right flank of his coalition, which has threatened to withdraw from government, something that could possibly lead to his ouster.

His far-right coalition partner and minister of national security, Itamar Ben-Gvir, has recently threatened to quit if Mr. Netanyahu negotiates a deal with Hamas that frees the hostages but allows the group to retain power.

“While claiming to defend Israel’s national security interests in Gaza, Netanyahu has a personal incentive to continue this war and expand it, if possible, knowing that as soon this war ends, his day of reckoning with the Israeli people will arrive and his career will be over,” said Nader Hashemi, a professor of Middle East politics at Georgetown University.

“He oversaw national security on Oct. 7, and Israelis will demand accountability for what happened,” said Mr. Hashemi, who, using a nickname for the prime minister, argued that “this is complicating negotiations between Blinken and Bibi and might derail the entire prospects for a cease-fire deal with Hamas.”

Mr. Netanyahu has rejected claims that he has allowed personal considerations to supersede Israeli interests. Asked for comment, his office said that his critics overseas “don’t realize that the P.M. reflects the view of most Israelis.”

Biden administration officials say the negotiations will continue in the days ahead, and Mr. Blinken has said he still believes there can be consensus. And despite Mr. Netanyahu’s comments the day earlier, Israeli officials on Thursday suggested that Israel was still open to negotiation.

“There is agreement among members of the governing coalition and particularly among individual members of the government that we do have to get the hostages back and to make a deal,” Miki Zohar, a government minister, said in a radio interview on Thursday morning.

“But not at any price,” Mr. Zohar said. “Stopping the war, for example, they won’t agree to.”

Israeli leaders believed that Hamas’s core demands were unacceptable, but there was room for discussion if the offer seemed like an opening bid, according to two Israeli government officials who spoke on the condition of anonymity to discuss a sensitive matter.

While Mr. Netanyahu may not be able to agree to a permanent halt in fighting or the release of all Palestinian prisoners, he might consent to free roughly 1,000 prisoners over a monthlong truce, said Nadav Shtrauchler, a political analyst who was once Mr. Netanyahu’s media strategist.

A longer truce of, as Hamas has proposed, could easily become permanent — something Mr. Netanyahu could not live with, said Mr. Shtrauchler.

“He still left a window open,” said Mr. Shtrauchler. “The door has been closed, but the window is still open. Not for that deal, which he couldn’t accept, but for a different deal.”

Israel began to bombard Gaza after Hamas attacked on Oct. 7, killing about 1,200 people, according to Israeli officials. A ground invasion soon followed and in four months of the military offensive, more than 27,000 people in Gaza — most of them women and minors — have been killed, according to the territory’s health officials.

“Netanyahu faces huge political constraints in agreeing to anything like what Hamas appears to want,” said Aaron David Miller, a longtime Middle East peace negotiator now at the Carnegie Endowment for International Peace.

“In fact, the only party that seems to be in a hurry is the Biden administration. They want to change the disastrous images in Gaza; relieve political pressure at home and try to wrap this up with an Israeli- Saudi deal. The problem is, based on my experience, Middle East negotiations have two speeds — slow and slower.”