By Nidal al-Mughrabi and Maayan Lubell

CAIRO/JERUSALEM (Reuters) -Israeli strikes killed at least 60 people across Gaza on Monday in some of the heaviest attacks in weeks as Israeli officials were due in Washington for a new ceasefire push by U.S. President Donald Trump.

A day after Trump called to "Make the deal in Gaza, get the hostages back", Israel's strategic affairs minister Ron Dermer, a confidant of Prime Minister Benjamin Netanyahu's, was travelling to Washington for talks on Iran and Gaza, according to an Israeli official and a source familiar with the matter.

Dermer was expected to begin meetings with Trump administration officials on Tuesday, the source in Washington said.

But on the ground in the Palestinian enclave, there was no sign of fighting letting up. The Israeli military issued evacuation orders on Monday to residents in large districts in the northern Gaza Strip, forcing a new wave of displacement.

"Explosions never stopped; they bombed schools and homes. It felt like earthquakes," said Salah, 60, a father of five children, from Gaza City. "In the news we hear a ceasefire is near, on the ground we see death and we hear explosions."

"Look at us, we are not just numbers and not just pictures. Every day martyrs like this," said displaced woman Amani Swalha, standing in the rubble of a Gaza city school hit in a strike. "It is our right to live, and to live with dignity, not like this in humiliation."

Israeli tanks pushed into the eastern areas of Zeitoun suburb in Gaza City and shelled several areas in the north, while aircraft bombed at least four schools after ordering hundreds of families sheltering inside to leave, residents said.

At least 58 people were killed in Israeli strikes on Monday, health authorities said, including 10 people killed in Zeitoun and at least 13 killed southwest of Gaza City. Medics said most of the 13 were hit by gunfire, but residents also reported an airstrike.

Twenty-two people, including women, children and a local journalist were killed in an Israeli airstrike on a beachfront cafe in Gaza City, medics said. The Palestinian Journalist Syndicate said more than 220 journalists had been killed in Gaza since the war began in October 2023.

The Israeli military said it struck militant targets in northern Gaza, including command and control centers, after taking steps to mitigate the risk of harming civilians.

There was no immediate word from Israel on the reported casualties southwest of the Gaza Strip and the beachfront cafe.

The bombardment followed new evacuation orders to vast areas in the north, where Israeli forces had operated before and left behind wide-scale destruction. The military ordered people there to head south, saying that it planned to fight Hamas militants operating in northern Gaza, including in the heart of Gaza City.

'MAKE THE DEAL'

Alongside talks on Gaza ceasefire prospects, Dermer also plans to discuss Netanyahu's possible visit to the White House in coming weeks, according to the source familiar with the matter.

In Israel, Netanyahu's security cabinet was expected to convene to discuss the next steps in Gaza.

On Friday, Israel's military chief said the present ground operation was close to having achieved its goals, and on Sunday, Netanyahu said new opportunities had opened up for recovering the hostages, 20 of whom are believed to still be alive.

Palestinian and Egyptian sources with knowledge of the latest ceasefire efforts said that mediators Qatar and Egypt have stepped up their contacts with the two warring sides, but that no date has been set yet for a new round of truce talks.

A Hamas official said that progress depends on Israel changing its position and agreeing to end the war and withdraw from Gaza. Israel says it can end the war only when Hamas is disarmed and dismantled. Hamas refuses to lay down its arms.

Israeli Foreign Minister Gideon Saar said Israel has agreed to a U.S.-proposed 60-day ceasefire and hostage deal, and put the onus on Hamas. He told reporters: "Israel is serious in its will to reach a hostage deal and ceasefire in Gaza."

Austrian Foreign Minister Beate Meinl-Reisinger, speaking in Jerusalem alongside her Israeli counterpart, said the humanitarian situation in Gaza was “unbearable.”

“The suffering of civilians is increasingly burdening Israel's relations with Europe. A ceasefire must be agreed upon,” she said, calling for the unconditional release of hostages by Hamas and for Israel to allow the uninterrupted flow of humanitarian aid into Gaza.

Israel says it continues to allow aid into Gaza and accuses Hamas of stealing it. The group denies that accusation and says Israel uses hunger as a weapon against the Gaza population.

The U.S. has proposed a 60-day ceasefire and the release of half the hostages in exchange for Palestinian prisoners and the remains of other Palestinians. Hamas would release the remaining hostages as part of a deal that guarantees ending the war.

The war began when Hamas fighters stormed into Israel on October 7 2023, killed 1,200 people, most of them civilians, and took 251 hostages back to Gaza in a surprise attack that led to Israel's single deadliest day.

Israel's subsequent military assault has killed more than 56,000 Palestinians, most of them civilians, according to the Gaza health ministry, displaced almost the whole 2.3 million population and plunged the enclave into a humanitarian crisis.

More than 80% of the territory is now an Israeli-militarized zone or under displacement orders, according to the U.N.

(Writing by Nidal al-Mughrabi and Maayan Lubell. Additional reporting by Alex Cornwell; Editing by William Maclean)