Gaza City residents fled under Israeli fire as tanks thrust deeper into the heart of the city on Tuesday, the second day of a stepped-up military offensive that Palestinian militant group Hamas said could jeopardise ceasefire talks.

Qatari and Egyptian mediators, backed by the United States, have accelerated their efforts this week to seal a ceasefire deal aimed at ending the Gaza war and releasing Israeli hostages in the enclave in return for Palestinian prisoners jailed in Israel.

On Tuesday, Israeli tanks deepened their incursion into some Gaza City districts including Shejaia, Sabra and Tel Al-Hawa, where residents reported the previous day some of the most fierce fighting since the start of the war between Israel and the Islamist Hamas.

Footage circulated on social media on Tuesday showed families packed onto donkey carts and in the backs of trucks piled with mattresses and other belongings racing through the city's streets to flee areas under Israeli evacuation orders.

"Gaza City is being wiped out, this is what is happening. Israel is forcing us to leave homes under fire," Um Tamer, a mother of seven, told Reuters via a chat app.

She said it was the seventh time her family had fled their house in Gaza City, in the north of the enclave and one of Israel's first targets at the start of the war in October.

"We can't take it anymore, enough of death and humiliation. End the war now," she said.

The Palestinian Red Crescent said all of its medical clinics were out of service in Gaza City due to the evacuation orders.

The assault has unfolded as senior U.S. officials were in the region pushing for a ceasefire after Hamas made concessions last week. But the renewed campaign "could bring the negotiation process back to square one," Hamas quoted leader Ismail Haniyeh as saying on Monday.

The armed wings of Hamas and the Islamic Jihad said they fought against Israeli forces in Tel Al-Hawa with anti-tank rockets and mortar fire, and inflicted casualties. There was no immediate response from the Israeli military on those claims.

In a statement, the Israeli military said troops were continuing operations in Gaza City, following intelligence indicating the presence of Hamas and Islamic Jihad in the area. It said Israeli forces had taken dozens of fighters out of action and located numerous weapons.