In Jabalia, tanks were trying to advance towards the heart of the camp on Monday, the biggest of Gaza's eight historic refugee camps. Residents said tank shells were landing at the centre of the camp and that air strikes had destroyed clusters of houses.
Residents and medics said several people were killed and wounded in a series of air strikes on the camp overnight. Medics said they have been unable to send teams to some of the bombed areas because of the intensity of the Israeli bombardment but they have reports of fatalities.
In Rafah, near the border with Egypt, Israel stepped up aerial and ground bombardments on the eastern areas of the city, killing people in an air strike on a house in the Brazil neighbourhood.
Residents said Israeli tanks have cut off the Salahuddin Road that bisects the eastern part of the city, while the eastern part of Rafah remained a "ghost town".
Intense fighting was reported and Israeli forces and tanks were seen in the southeast area of Rafah, residents said.
Jack Lew, the United States' ambassador to Israel, signalled on Sunday that the Rafah incursion was still on a scale acceptable to Washington.
Lew referred to a disclosure by US President Joe Biden to CNN last week that a bomb shipment to Israel was on hold as a warning not to "go into Rafah".
"The president was clear in the interview he gave the other evening that what Israel has done so far hasn't crossed over into the area where our disagreements lie," Lew told Israel's Channel 12 TV, without elaborating on what that area entails.
"I'm hoping we don't end up with real disagreement."
Hamas' armed wing said its fighters were engaged in gun battles with Israeli forces in one of the streets east of Rafah, and in the east of Jabalia.
In Israel, the military sounded sirens several times in areas near Gaza, warning of potential Palestinian cross-border rocket and or mortar launches.
Late on Saturday, the Israeli military said forces operating in Jabalia were preventing Hamas, which rules Gaza, from re-establishing its military capabilities there.
"They were bombing everywhere, including near schools that are housing people who lost their houses," Jabalia resident Saed, 45, told Reuters via a chat app on Sunday.
"War is restarting, this is how it looks in Jabalia."
The death toll in Israel's military operation in Gaza has now passed at least 35,000 Palestinians, according to Gaza's health ministry. The bombardment has laid waste to the coastal enclave and caused a deep humanitarian crisis.
The war was triggered by a Hamas-led attack on southern Israel on October 7 last year in which some 1200 people were killed and more than 250 people taken hostage, according to Israeli tallies.
Israel says 620 soldiers have been killed in the fighting, more than half of them during the initial Hamas assault.
