Secretary of State Antony J. Blinken was heading to Israel on Sunday to try to clinch a deal that could end the war in Gaza, even as the Middle East remained on edge amid the looming threat of wider regional conflict.

The visit, part of an intensive diplomatic campaign led by the Biden administration, comes days after Israel’s negotiating team held talks in Qatar with senior American officials, as well as Qatari and Egyptian representatives who are mediating between Israel and Hamas.

Those talks ended without a major breakthrough, but the White House said in a statement on Friday that the United States had put forward a “bridging proposal,” with Egyptian and Qatari support, intended to close remaining gaps between the sides. It said teams would continue to hash out details for implementing the deal and that senior negotiators hoped to reconvene in Cairo before the end of this week to finalize an agreement.

In a statement issued on Saturday night, after the sabbath, the office of Israel’s prime minister, Benjamin Netanyahu, said the Israeli negotiating team had expressed “cautious optimism” over the possibility of advancing toward a deal based on the bridging proposal. It did not offer further details, and portrayed Hamas as the obstacle to reaching an agreement.

Mr. Blinken is scheduled to meet on Monday with Mr. Netanyahu, whom some officials have accused of stalling by adding new conditions for a deal.

The potential deal would be carried out in three phases and is based on principles that were laid out by President Biden on May 31 and subsequently endorsed by the United Nations Security Council. It would usher in a cease-fire in Gaza and involve the release of the hostages being held captive in the enclave in exchange for Palestinian prisoners and detainees held in Israel.

The Biden administration has created a degree of linkage between the cease-fire efforts and the threat of Iranian-led retaliation against Israel for the back-to-back assassinations of senior figures from Hezbollah, the Iranian-backed Lebanese militia, and Hamas in Beirut and Tehran in late July.

Amid fears that any reprisals and subsequent Israeli counterstrikes could escalate into a broader regional war, American officials have expressed hope that progress on the diplomatic front could stave off a broader conflagration.

In the meantime, the fighting in Gaza has continued. The Israeli military said on Sunday that its troops were operating in the central and southern Gaza Strip.