Mediators plan to move ahead with a summit next week pursuing a cease-fire agreement in Gaza, Israeli officials said on Friday, after Israeli security chiefs sought to obtain Egyptian consent for a postwar Israeli presence along Gaza’s border with Egypt.

The issue of Israeli troops on the border has emerged as a particularly contentious dispute in the overall negotiations for a cease-fire deal between Israel and Hamas, talks that mediators from Egypt, Qatar and the United States have struggled for months to keep alive.

Hamas has repeatedly rejected the idea of an Israeli presence in the border area, saying that any deal to stop the war must involve Israel’s complete withdrawal from Gaza. Prime Minister Benjamin Netanyahu of Israel has argued that the tunnels in the area, known as the Philadelphi Corridor, have served as a major conduit for smuggling weapons into Gaza, and that abandoning them would allow Hamas to quickly rearm.

Egypt, as a neighboring country and a mediator in the truce talks with a significant stake in the war’s outcome, is also key to reaching a truce agreement. The government has said that keeping Israeli troops at its Gaza border could raise national security concerns and potentially threaten Egyptian-Israeli relations. Egypt also says it has already taken aggressive action to destroy tunnels and stop smuggling.

Faced with an apparent impasse, diplomats have tried to push toward some kind of agreement, veering for weeks between tentative optimism and deadlock while saying little about the talks in public. Both Israeli and Hamas officials have blamed each other for the failure to reach a deal, which also aims to free more than 100 hostages held in Gaza.

President Biden on Friday afternoon spoke by telephone with Qatar’s emir, Sheikh Tamim Bin Hamad Al-Thani, and made a separate call to President Abdel Fattah el-Sisi of Egypt, as he continued his effort to find a formula that would lead to a cease-fire. Earlier in the week he called the Israeli prime minister.

On Thursday, Israeli security chiefs traveled briefly to Cairo to continue talks with Egyptian mediators. American officials, including the C.I.A. director, William J. Burns, and President Biden’s Middle East envoy, Brett McGurk, also took part, a White House official said on Friday.

“The process is moving forward,” John F. Kirby, a White House national security spokesman, told reporters on Friday. “It’s moving forward in the way we had outlined earlier.”

The talks came after Secretary of State Antony J. Blinken visited Israel, Egypt and Qatar this week to push for an American proposal intended to bridge the differences between Israel and Hamas over a cease-fire deal. Mr. Blinken declared that Israel had accepted the plan, the details of which have not been made public, and that it was now up to Hamas to accept the deal. But Hamas has rejected that characterization.

At the latest Cairo meeting, David Barnea, the head of the Mossad intelligence agency, presented new maps showing the possible redeployment of Israeli forces along the Philadelphi Corridor, according to two Israeli officials familiar with the matter and who spoke on condition of anonymity as they were not authorized to speak publicly. They did not provide further details on the proposal.

Egyptian officials have not commented publicly on the meeting, but Egypt’s position on the corridor has been clear: It has consistently said that the longstanding “security agreements and protocols” the countries have signed to govern the area commit Israel to keeping troops away from it.

Israel and Hamas have been negotiating for months on the basis of a three-stage cease-fire framework endorsed by Mr. Biden and the United Nations Security Council. The agreement stipulates an initial truce — during which hostages would be swapped for Palestinians jailed in Israel — that would lead to a permanent cease-fire.

U.S., Egyptian and Qatari mediators are urgently pushing for a deal in the hope that it will help avert a wider regional conflict after two killings last month that were widely attributed to Israel. In one, an explosion killed Hamas’s political leader, Ismail Haniyeh, in Tehran; hours earlier, an Israeli airstrike in the southern suburbs of Beirut killed a senior commander of Hezbollah, the Iran-backed Lebanese group. Iran and its allies have vowed to retaliate against Israel for both killings.

Israeli forces and Hezbollah militants have been adversaries for years, but cross-border shelling, missiles and drone attacks have intensified since the start of the war in Gaza, fueling fears of a wider war. Lebanon’s Health Ministry said on Friday that two people had been killed, including a 7-year-old, in an Israeli drone strike in southern Lebanon.

The hostilities along Israel’s northern border have displaced tens of thousands of people and caused deaths and injuries on both sides. About 500 people, including at least 100 civilians, have been killed in Lebanon over the past 10 months, according to the U.N. and Lebanon’s Health Ministry. Hezbollah attacks since the start of the war have killed around 50 people in Israel, more than half of them civilians, according to the Israeli prime minister’s office.

U.S. officials are hoping a truce in Gaza would help calm tensions across the region. They have also increased pressure on Israel to curb violence perpetrated by Jewish settlers in the Israeli-controlled West Bank, another regional flashpoint where conflict has grown since the war began.

The lawyers of three Israeli settlers said on Friday that their clients had been detained after being accused of taking part in an attack on a Palestinian village in the West Bank in which a 23-year-old Palestinian man was killed and homes were set on fire.

The Israeli Defense Ministry has placed the three settlers, whose identities were redacted in warrants shared by their lawyers, under administrative detention — imprisonment without charges or trial that has primarily been used against Palestinians in the West Bank. The details of the settlers’ involvement in the attack were not clear.

Dozens of Israeli Jewish settlers rampaged through the Palestinian village of Jit on the night of Aug. 15, dressed in dark clothes and armed with rocks and assault rifles, witnesses said. Some of the attackers wore masks.

One of the attackers shot and killed Rasheed al-Seda, 23, who had joined residents trying to defend the village while armed with little more than stones, according to residents and Palestinian health officials. Settlers also set fire to four houses and six vehicles, according to Yesh Din, an Israeli rights group.

Israeli rights activists say that the Israeli military generally avoids confronting violent settlers as a matter of policy, and soldiers routinely enable settler violence against Palestinians and their property, sometimes watching from the sidelines.

The attack in Jit has underlined divisions in the Israeli governing coalition. Residents of Jit say top officials in Mr. Netanyahu’s government, notably Itamar Ben-Gvir, the far-right minister of national security, bore responsibility for the Aug. 15 attack because of their inflammatory rhetoric. Mr. Ben-Gvir also eased gun-control regulations for Israeli civilians in the wake of the Hamas-led Oct. 7 attack on Israel.

On Friday, Israel’s defense minister, Yoav Gallant, harshly criticized Mr. Ben-Gvir on social media after Ronen Bar, the head of the country’s internal security service, Shin Bet, reportedly accused Mr. Ben-Gvir in a letter of encouraging Jewish extremists and helping to cause “indescribable damage” to Israel.

Mr. Gallant accused Mr. Ben-Gvir of “reckless actions that endanger Israel’s national security and create internal division in the nation.”

Reporting was contributed by Vivian Yee, Emad Mekay, Eric Schmitt, Johnatan Reiss and Thomas Fuller.