Israeli tanks, under cover from heavy fire from air and ground, pushed further into Jabalia in the northern Gaza Strip on Monday, residents and Hamas media said, while airstrikes hammered Rafah in the south.
In Jabalia, tanks were trying to advance towards the heart of the camp, 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.
Israeli troops forced hundreds of Palestinians housed in shelters to leave.
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 are now stationed east of the Salahuddin Road that bisects the eastern part of the city, with the highway cut off by intense fighting. Residents added the eastern part of Rafah remained a "ghost town".
Hamas said their 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.
The army sent tanks back into Zeitoun, as well as Al-Sabra, where residents also reported heavy bombardments that destroyed several houses, including high-rise residential buildings.
US Secretary of State Antony Blinken on Sunday warned that Israel was risking facing an insurgency in Gaza without a post-war plan for the enclave.
Israeli Defence Minister Yoav Gallant office said on Monday he'd briefed Blinken about Israel's military operation in the southern Gazan city of Rafah and takeover of the nearby border crossing into Egypt.
In an overnight phone call, Gallant and Blinken "discussed developments in Gaza...and the precise operation in the Rafah area against remaining Hamas battalions, while securing the crossing," the statement said.
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.