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"We want to get these hostages out, we want to get a ceasefire in place for six weeks, we want to increase humanitarian assistance," White House national security spokesperson John Kirby said, adding that reaching an agreement would be the "absolute best outcome".
More than 34,600 Palestinians have been killed in the conflict, according to Gaza health officials.
The UN has said that famine is imminent in the enclave.
Any truce would be the first pause in fighting since a week-long ceasefire in November, during which Hamas freed around half of the hostages.
Since then, all efforts to reach a new truce have foundered over Israel's insistence that it would discuss only a temporary pause.
Taher al-Nono, a Hamas official and adviser to Haniyeh, told Reuters that the proposal met the group's demands for reconstruction efforts in Gaza, return of displaced Palestinians and a swap of Israeli hostages for Palestinian prisoners in Israeli jails.
The Hamas deputy chief in Gaza, Khalil al-Hayya, told Al Jazeera television that the proposal comprised three phases, each of six weeks, with Israel to pull its troops out of Gaza in the second phase.
Earlier yesterday, Israel ordered the evacuation of parts of Rafah, the city on the Egyptian bordered that has served as the last sanctuary for around half of Gaza's 2.3mn residents.
Israel's closest ally, the United States, has called on it not to assault Rafah, saying that it must not do so without a full plan in place to protect civilians there, which has yet to be presented.
Washington is committed to stopping Israel's attack on Rafah, the US official said.
Israel said yesterday that it was conducting limited operations on the eastern part of Rafah.
That was being accompanied by massive air strikes, according to Palestinian residents.
"They have been firing since last night and today after the evacuation orders, the bombardment became more intense because they want to frighten us to leave," Jaber Abu Nazly, a 40-year-old father of two, told Reuters via a chat app. "Others are wondering whether there is any place safe in the whole of Gaza."
Instructed by Arabic text messages, phone calls, and flyers to move to what the Israeli military called an "expanded humanitarian zone" around 12 miles away, some Palestinian families began trundling away in chilly spring rain.
Some piled children and possessions onto donkey carts, while others left by pick-up or on foot through muddy streets.
As families dismantled tents and folded belongings, Abdullah al-Najar said that this was the fourth time he had been displaced since the fighting began seven months ago.
"God knows where we will go now. We have not decided yet."