Fierce fighting has been reported in Jabalia in northern Gaza, after the Israeli military went back into areas where it said Hamas had regrouped.
Residents who fled said they had seen tanks advancing towards Jabalia’s refugee camp, which has come under heavy bombardment since Saturday.
Palestinian armed groups also said they were battling troops in the camp.
Meanwhile, the UN has said 360,000 people have fled Rafah, in the south, since an offensive began a week ago.
The Israeli military has ordered the evacuation of the eastern third of the city, which is swollen with more than a million Palestinians taking refuge.
On Sunday, US Secretary of State Antony Blinken warned Israel that a full-scale offensive in Rafah might provoke “anarchy” without eliminating Hamas.
His comments echoed briefings to Israeli media by unnamed senior Israeli military figures, who said Hamas’s resurgence in northern Gaza was due to a lack of a specific plan from Israel’s government for the “day after” the war.
The military scaled down operations in the north in January after declaring that it had “dismantled” Hamas’s battalions there. But that left a power vacuum in which the group has been able to rebuild.
An estimated 300,000 people trapped in the devastated region are also experiencing a “full-blown famine” due to a lack of aid deliveries, according to the head of the World Food Programme.
Residents who were filmed fleeing Jabalia on foot on Monday morning said they had decided to leave after seeing tanks advance into the area.
“We don’t know where to go. We have been displaced from one place to the next,” one woman told Reuters news agency. “We are running in the streets. I saw it with my own eyes. I saw the tank and the bulldozer.”
The military wings of Hamas and Palestinian Islamic Jihad – which are both proscribed as terrorist organisations by Israel, the UK, US and other countries – said their fighters were targeting Israeli forces in and around Jabalia camp with mortars, anti-tank missiles and machine-guns.
Hamas-affiliated Safa news agency also reported clashes between Palestinian armed groups and Israeli tanks east of Jabalia camp’s market, near several UN-run schools which were being used as shelters by civilians.
The official Palestinian news agency Wafa meanwhile reported that two people were killed in Israeli strikes on homes in Jabalia camp on Monday and that a number of other people were killed in a strike in Jabalia town.
It also cited ambulance crews as saying the bodies of 20 civilians had so far been recovered in Jabalia and taken to Kamal Adwan hospital in Beit Lahia.
There was no immediate comment from the Israel Defense Forces (IDF).
On Sunday, the IDF said troops had begun an operation in Jabalia the previous evening “based on intelligence information regarding attempts by Hamas to reassemble its terrorist infrastructure and operatives in the area”.
It came after civilians there were told to evacuate to western Gaza City.
The IDF also said it was operating in the Zeitoun area, in eastern Gaza City, to “eliminate terrorists and dismantle terrorist infrastructure”.
Safa reported that Zeitoun was hit by Israeli shellfire on Monday morning.
The UN agency for Palestinian refugees, Unrwa, said the bombardment and evacuation orders had “created more displacement and fear for thousands of families” in northern Gaza.
Unrwa also warned that the civilians affected, like the estimated 360,000 who have fled Rafah in the past week, had “nowhere safe to go”.
After seven months of war in Gaza, Israel has insisted victory is impossible without taking Rafah and eliminating the last remaining Hamas battalions. But UN and Western powers have warned that an all-out assault could lead to mass civilian casualties and a humanitarian catastrophe.
There have been desperate scenes in Rafah since the IDF said last Monday that it was beginning a “precise operation against Hamas” in eastern Rafah.
On Saturday, flyers were dropped ordering residents to evacuate more eastern neighbourhoods, including those close to the city centre.
Unrwa’s deputy Gaza director Scott Anderson, who is based in western Rafah, told the BBC on Monday that the Israeli operation now extended “about a third of the way across Rafah” and encompassed the old town.
He said he could hear explosions and air strikes, and that the fighting had affected the operations of hospitals and many of Unrwa’s facilities in Rafah refugee camp, including a primary healthcare facility.
Palestinian media reported that four people, including a child, were killed on Monday in an Israeli air strike on a home in the Brazil neighbourhood, which is just south-east of the camp.
Hamas’s military wing also said it had targeted Israeli forces east of Rafah.
Many people sheltering in central and western areas that are not subject to evacuation orders have also been leaving out of fear that the Israeli operation will soon expand into a large-scale offensive.
Ghada el-Kurd, a mother-of-two, told the BBC on Monday that she had recently fled to Deir al-Balah in Gaza’s Middle Area – the seventh time she has been displaced during the war.
She said the streets of Deir al-Balah were “full of sewage” and that the destruction of homes and infrastructure was “huge, not like Rafah”.
She added: “I’m lucky enough to have a home, but most other people are living in tents and are [suffering from] the heat, a lack of water and a lack of food.”
Deir al-Balah is part of the IDF-designated “expanded humanitarian area”, which stretches north from the coastal strip of al-Mawasi to the city of Khan Younis and Gaza’s Middle Area.
The IDF has told evacuees they will find field hospitals, tents and aid supplies there, but Mr Anderson of Unrwa said it lacked the necessary infrastructure.
In al-Mawasi, he explained, “everything that people have access to has to be trucked in… that includes food, water, moving solid waste”.
“For people who are displacing to Khan Younis it is very similar… That is still recovering from the Israeli operation that took place a few months ago.”
Mr Anderson also appealed to Israel and Hamas to agree to the creation of “fixed corridor” that would allow UN lorries to move safely in and out of the nearby Kerem Shalom border crossing with Israel – a key entry point for aid.
Israel said it had reopened the crossing last Wednesday after closing it for four days due to Hamas rocket fire. But the UN said it was too dangerous to collect aid because of the hostilities in eastern Rafah.
The Rafah border crossing with Egypt has also been closed since Israeli troops took full control of the Palestinian side last Tuesday.
In a statement issued on Sunday, Hamas accused Israel of “escalating their brutal massacres in various areas of the Gaza Strip”.
Israel launched a military campaign in Gaza to destroy Hamas in response to the group’s cross-border attack on southern Israel on 7 October, during which about 1,200 people were killed and 252 others were taken hostage.
More than 35,090 people have been killed in Gaza since then, according to the territory’s Hamas-run health ministry.