This week, the White House dispatched a top aide to the president, Brett McGurk, to Israel for discussions with the government there, while William J. Burns, the C.I.A. director, traveled to Doha, Qatar, for talks with the Egyptian, Qatari and Israeli officials negotiating over the release of the hostages.

On Friday, Mr. McGurk will lead the American delegation for further talks in Cairo on the framework of a three-phase deal that is backed by the United States and the United Nations. After holding meetings with the Israeli negotiating team on Thursday, Mr. Netanyahu dispatched a delegation led by the head of Israel’s Shin Bet intelligence service to Cairo for continued discussions.

Negotiators have tried to overcome hurdles to a deal by reaching precise agreements on the exchanges of Palestinian prisoners for Israeli hostages in its first phase. At the same time they have pushed for agreement on the broader framework for subsequent phases of the deal.

The framework discussions include the two most contentious issues: whether Israel will agree to end the war, withdraw from Gaza and respect a permanent cease-fire; and whether Hamas will agree to give up control of the Gaza Strip, according to a person briefed on the negotiations.

Both Israel and Hamas remain exceptionally wary about whether the other side is truly ready to make concessions.

Husam Badran, a senior Hamas official, said his group had shown “great flexibility” in discussions with mediators, especially in making language changes, but had held firm to its demand that Israel agree to a permanent cease-fire.

“We’re not obstinate and rigid in negotiating,” he said in an interview in Doha. “If there are some phrases that will make the negotiations easier and lead to the same result — the end of the war — we have no problem.”

Palestinians have grown weary of the war, which has killed tens of thousands of people in Gaza. While most still blame Israel for the death and destruction, anger at Hamas — and a willingness to express that resentment — is growing.

One reason to be careful, those researchers said, is that any estimate of excess deaths would rely on data from Gaza’s health sector, which has been devastated by the conflict. Another reason, they said, is that it is hard to predict how epidemics and hunger, two threats to human life that can be triggered by war, will evolve. And Israel has not permitted researchers to enter the enclave since the start of the war last October.

The letter in The Lancet, which said that counting indirect deaths in Gaza was “difficult but essential,” based its estimate on looking at previous studies of recent conflicts, which indicated that three to 15 times as many people died indirectly for every person who had died violently. Applying what they called a “conservative estimate of four indirect deaths per one direct death,” the authors wrote that it was “not implausible” to estimate that about 186,000 deaths could eventually be attributable to the conflict in Gaza.

The letter, which The Lancet said had not been peer-reviewed, as is the case with other letters it publishes, provoked a significant response. The Board of Deputies of British Jews, which represents the Jewish community in Britain, said that the estimate was “little more than conjecture.”

Col. Elad Goren, an official with COGAT, the arm of the Israeli military that implements policy in Gaza, sidestepped a question about excess deaths.

Salim Yusuf, a cardiologist and epidemiologist in Canada who co-wrote the letter, said in an email that the estimate was based on studies of past conflicts and acknowledged that, “inevitably, these are projections.” “The point is that the real numbers of dead will be very large,” he said.

Michael Spagat, a professor of economics at Royal Holloway College at the University of London, who has written about the toll of the war in Gaza, wrote in an analysis that the letter “lacks a solid foundation and is implausible.” He argued that the authors had compared Gaza with a small and unrepresentative sample of other conflicts, and that conditions in Gaza, a small territory under intense international attention, are unique.

In an interview, Mr. Spagat cited other reasons to be cautious when discussing excess deaths in Gaza. He said that fears of major outbreaks of infectious diseases such as cholera have yet to materialize and that, although humanitarian agencies are warning of catastrophic levels of hunger, there is little evidence of widespread deaths because of starvation.

Still, Mr. Spagat said that it was “fair to call attention to the fact that not all of the deaths are going to be direct violent ones.”

The letter in The Lancet is not the first effort to quantify the human toll in Gaza beyond the figures reported by Gazan health authorities.

In February, epidemiologists at Johns Hopkins University and the London School of Hygiene and Tropical Medicine produced a model showing three different war scenarios affecting overall deaths in Gaza. They projected that if fighting and humanitarian access remained at the same levels, there could be an additional 58,260 deaths in the six months from March through August. Around 9,000 deaths have been directly attributed to the war since then by Gaza’s health ministry.

The health ministry says that more than 38,000 people have died in Gaza since Oct. 7, when Hamas, which controls the territory, led an attack on Israel in which 1,200 people were killed. While the ministry’s tally is broadly accepted, there remain questions about its methodologies and record keeping, as well as contradictions between its statements and underlying data. Most civilian victims, the ministry says, are women and children. But the figures do not distinguish between civilians and combatants.

The subject of excess deaths is sensitive because it touches on the collateral cost of Israel’s war against Hamas. On top of the large death toll, the attacks have damaged hospitals and shelters. Aid officials say that Israel has also restricted access to the fuel that medical facilities need to operate. Israeli officials say they do all they can to spare civilians, but blame Hamas for placing its forces in urban centers and civilian facilities. They have also said that aid agencies’ logistical difficulties, rather than Israeli restrictions, are to blame for the limited amount of humanitarian aid that is getting to Gazans.

Before the war, Gaza’s health sector produced reliable data, which helps in modeling excess deaths, but lack of access to Gaza for researchers makes the task more difficult, according to Zeina Jamaluddine, an epidemiologist at the London School of Hygiene and Tropical Medicine.

Gabby Sobelman contributed reporting.