Hamas has said it has accepted a Gaza ceasefire proposal from Egypt and Qatar.
The Islamist faction said in a statement that its chief, Ismail Haniyeh, had informed Qatar's prime minister and Egypt's intelligence chief of its acceptance of their proposal.
Senior Hamas member Khalil al-Hayya said a proposal agreed by the group includes a three-stage truce with the goal of a lasting ceasefire.
Hayya told Qatar-based Al Jazeera channel that each phase would last 42 days, and the deal includes plans for a complete Israeli withdrawal from Gaza, the return of Palestinians displaced by the ongoing war, an exchange of hostages and prisoners and the aim of a "permanent ceasefire".
A senior Hamas official said Israel must now decide whether it accepts or "obstructs" a truce in Gaza.
"After Hamas agreed to the mediators' proposal for a ceasefire, the ball is now in the court of Israeli occupation, whether it will agree to the ceasefire agreement or obstruct it," the official told AFP on condition of anonymity.
Israeli Prime Minister Benjamin Netanyahu's office said later that Hamas' latest truce proposal falls short of Israel's demands but Israel would send a delegation to meet with negotiators to try to reach an agreement.
An Israeli official said the proposal that Hamas had accepted was a "softened" version of an Egyptian proposal, which included "far-reaching" conclusions that Israel could not accept.
"This would appear to be a ruse intended to make Israel look like the side refusing a deal," said the Israeli official, who spoke on condition of anonymity.
'Return of all the hostages'
Families of hostages held in Gaza have demanded that Israel seize the "opportunity" provided by Hamas's approval of a truce proposal to strike a deal for the return of all the captives.
The Palestinian militant group's announcement on Monday "must pave the way for the return of the ... hostages held captive by Hamas for the past seven months," the Hostage Families and Missing Families Forum said in a statement, insisting that "now is the time for all that are involved to fulfil their commitment and turn this opportunity into a deal for the return of all the hostages".
Rafah assault
Israel's army has reiterated its call for evacuations in Rafah as it prepares for a "ground operation" in the southern Gaza city.
"We also call on residents this evening to evacuate," military spokesman Daniel Hagari said in a broadcast address after the Hamas announcement, adding that throughout the day Monday, "air force aircraft targeted more than 50 terror targets in the Rafah area".
Earlier, US President Joe Biden "reiterated his clear position" to Prime Minister Benjamin Netanyahu after Israel defied US warnings over an assault on Rafah, the White House said.
Mr Netanyahu "agreed to ensure the Kerem Shalom crossing is open for humanitarian assistance for those in need," it added in a readout of their call, after Israel closed the key Gaza border crossing following a Hamas rocket attack.
Instructed by Arabic text messages, telephone calls, and flyers to move to what the Israeli military called an "expanded humanitarian zone" 20km away, some Palestinian families lumbered out under chilly spring rain, witnesses said.
A senior official of Hamas, the militant Palestinian group that governs Gaza, said the evacuation order was a "dangerous escalation that will have consequences".
"The US administration, alongside the occupation, bears responsibility for this terrorism," the official, Sami Abu Zuhri, said, referring to Israel's alliance with Washington.
Israel's military said it had begun encouraging residents of Rafah to evacuate in a "limited scope" operation.
It gave no specific reasons, nor did it say if any offensive action might follow.
"It has been raining heavily and we don't know where to go. I have been worried that this day may come, I have now to see where I can take my family," one refugee in Rafah, said.
Witnesses said the areas in and around Rafah to which Israel wants to move people are already crowded and there is almost no room for more tents to be added.
An Israeli offensive in Rafah "would be devastating for 1.4 million people" sheltering there, the UN agency for Palestinian refugees, UNRWA, said on X, adding it would keep a presence in Rafah as long as possible to provide aid.
Seven months into its war against Hamas, Israel has been threatening to launch incursions in Rafah, which it says harbours thousands of Hamas fighters and potentially dozens of hostages. Victory is impossible without taking Rafah, it says.
The prospect of a high-casualty operation worries Western powers and neighbouring Egypt.
The Rafah plan has opened an unusually public rift between Israel and Washington.
Speaking to his US counterpart, Israeli Defence Minister Yoav Gallant linked today's operation to the deadlock in indirect diplomacy, which he blamed on Hamas.
"During their discussion, Gallant discussed the efforts undertaken to achieve the release of hostages and indicated that at this stage, Hamas refuses the frameworks at hand," the Israeli Defence Ministry said in a statement.
"Gallant emphasised that military action is required, including in the area of Rafah, at the lack of an alternative," it added.
An Israeli broadcaster, Army Radio, said evacuations were focused on a few peripheral districts of Rafah, from which evacuees would be directed to tent cities in nearby Khan Younisand Al Muwassi.
Many residents in Rafah said they had received telephone calls to evacuate their homes in the targeted area, in line with the army announcement.
In an overnight aerial attack on Rafah, Israeli planes hit ten houses, killing 20 people and wounding several, medical officials said.
Three Israeli soldiers were killed yesterday in a Hamas rocket attack near Rafah, at the Kerem Shalom crossing into Gaza, while Palestinian health officials said at least 19 people were killed by Israeli fire.
Yesterday's crossing attack came as hopes dimmed for ceasefire talks in Cairo, with Hamas reiterating its demand for an end to the war in exchange for the freeing of hostages, and Israeli Prime Minister Benjamin Netanyahu flatly ruling that out.
"Our just war in Gaza continues with the exact same goals: the release of all hostages and the defeat of Hamas," Israeli Foreign Minister Israel Katz said on X, blaming the Palestinian group for the lack of progress in the Cairo talks.
The war began after Hamas stunned Israel with a cross-border raid on 7 October in which 1,200 people were killed and 252 hostages taken, according to Israeli tallies.
More than 34,600 Palestinians have been killed, 29 of them in the past 24 hours, and more than 77,000 have been wounded in Israel's assault, according to Gaza's health ministry.
Yesterday, a top UN official accused Israel of continuing to deny the United Nations humanitarian access in Gaza, where the UN food chief warned a "full-blown famine" has taken hold in the north of the enclave of 2.3 million people.
While not a formal declaration, World Food Programme Executive Director Cindy McCain said, in an NBC News interview broadcast yesterday, that based on the "horror" on the ground: "There is famine, full-blown famine, in the north, and it's moving its way south."
