Like many other Gazans, Mr. Skeik said that he had grown frustrated after several rounds of cease-fire negotiations fell through in the past. Previous American, Qatari and Egyptian efforts to bring both sides to an agreement have faltered, with Mr. Biden suggesting in February that a cease-fire was imminent, even as Hamas and Israel continued to remain far apart.

“The United States used to have a strong word when it wanted to stop any crisis in the world,” he said. “But nowadays, I see a different thing.”

The first phase of the proposal laid out by Biden calls for both sides to observe a temporary six-week cease-fire, while continuing to negotiate to reach a permanent one. That scared Mr. Skeik, who said that without an immediate permanent cease-fire, he was worried the fighting would continue after or even during the first phase.

“I want to get back to my old life,” he said from a cafe where he can connect to the internet. But Mr. Skeik was worried that Hamas would nitpick the language and drag out negotiations, which would further forestall the possibility of him going home.

“We want Hamas to sign this deal to maintain a long-term peace and cease-fire for us and our children to live in peace and safety,” said Anas al-Borno, a 36-year-old businessman from Gaza City who was displaced with his family to Deir al-Balah. But he was “still hopeless and pessimistic,” that Israel and Hamas would both agree to the deal, he added.

Some praised Mr. Biden for his speech last week, in which the president laid out details of the Israeli plan. It was an unusual move to speak for another country, and appeared to be a move to further pressure Mr. Netanyahu after months of American admonitions.

“I think what Biden said on TV was a sudden change for me and many other people,” said Ahmed al-Masri, a 21-year-old dentistry student from Gaza City. “The United States has chosen the route of surprises recently so I hope this comes true and is real,” he added.

But others doubted it would mean much.

“The United States must impose solutions to all sides, not just propose and suggest ideas,” said Raed al-Kelani, 47, a civil servant from northern Gaza. He added that although he believed President Biden could press both Hamas and Mr. Netanyahu to agree to the deal, he was “only 50 percent optimistic.”