Israeli warplanes struck near Beirut for the first time in several days and airstrikes killed at least 52 people in central Lebanon on Friday, according to the Lebanese authorities, as diplomatic efforts to reduce the fighting between Israel and Hezbollah showed no sign of success.

Negotiations to reach a temporary truce between Israel and Hamas, Hezbollah’s ally in the Gaza Strip, also hit another impasse on Friday. Hamas appeared to rule out the possibility of a limited cease-fire in order to exchange hostages held in Gaza and Palestinian prisoners, according to a statement released through the group’s official media.

The Biden administration had sent key envoys including the C.I.A. director to the region this week, in a push to at least generate some momentum in talks to end Israel’s war against Hamas in Gaza and its spiraling conflict with Hezbollah in Lebanon.

But the envoys departed the region on both fronts without any apparent or immediate results in either conflict.

The military campaign Israel has waged in retaliation for Hamas’s Oct. 7 attacks has displaced some 90 percent of Gaza’s two million residents, razed swaths of the enclave’s cities and killed 39,000 people, according to Gaza’s health ministry, which does not distinguish between civilians and combatants.

Despite that, Hamas not only remains operational, but is recruiting new fighters both in Gaza and beyond, local residents and analysts say. Militants have also begun to re-emerge in areas that Israel had driven them out of months before.

For Hamas, the logic of insurgency means that simply surviving in the face of a far more powerful military provides a symbolic victory. With that comes a chance at staying power that outlasts any pain Israel has inflicted.

On Wednesday, Israel’s military said that a strike it conducted on July 13 had killed Muhammad Deif, the head of Hamas’s military wing, who is seen as an architect of the Oct. 7 attack on Israel. Hamas has yet to confirm the killing. Mr. Deif’s death, however, would represent the end of a yearslong Israeli effort to kill the man who is effectively the second-most senior leader after Israel’s most-wanted man, Yahya Sinwar, the head of Hamas in Gaza.

Israel gave final approval on Wednesday to a settlement project in the heart of the occupied West Bank that supporters and critics alike say will deal a major blow to the contiguity of territory that Palestinians hope will be part of a future independent state.

The project, known as E1, was delayed for more than two decades, often following pressure by the United States. But the Trump administration has been far less critical of settlements than previous administrations, or most of the international community, which generally considers them to be illegal and obstacles to Israeli-Palestinian peace.

Roughly 500,000 Israeli settlers and about 3 million Palestinians live in the West Bank, where the Israeli military holds overriding control. Israeli troops regularly raid Palestinian cities and restrict the movement of Palestinians. While Israeli settlers who live in the West Bank can vote in Israeli elections, their Palestinian neighbors have no say in them.

Under the plan, around 3,400 additional settlement housing units would be built on one of the most sensitive tracts of real estate in the West Bank: E1, short for East One. The plot lies just east of Jerusalem, which both Israelis and Palestinians consider their capital.

When President Trump hosted Prime Minister Benjamin Netanyahu of Israel at the White House in early April, a reporter reminded Mr. Trump that his 2024 campaign promise to end the war in Gaza remained unfulfilled.

Israel had recently broken a tenuous cease-fire in its 18-month war with Hamas and renewed its bombardment of Gaza. But Mr. Trump professed optimism.

“I’d like to see the war stop,” he replied. “And I think the war will stop at some point that won’t be in the too-distant future.”

One month later, prospects for peace in Gaza have dimmed even further.

Mr. Netanyahu warned on Monday of an “intensive” Israeli escalation in the Palestinian enclave after his security cabinet approved plans to call up tens of thousands of reservists for a fresh assault there.

A far-right Israeli politician said on Monday that saving the hostages in Gaza was not Israel’s “most important goal” in its war with Hamas, further stoking the debate in Israel over its objectives for the war.

Bezalel Smotrich, the country’s powerful finance minister, suggested in a radio interview that ensuring that Hamas no longer ruled the Gaza Strip after its deadly 2023 attack in southern Israel was a higher priority.

“We have promised the Israeli people that at the end of the war, Gaza will no longer be a threat to Israel,” said Mr. Smotrich, who has called for building Jewish settlements in the Palestinian enclave. “We need to eliminate the problem of Gaza.”

Israel launched the war in Gaza after the Hamas-led attack with at least two aims: One was to destroy Hamas and restore Israelis’ sense of safety after roughly 1,200 people were killed in the surprise attack. Another was to bring back the more than 250 people captured in the assault.