Hamas has fired rockets on the southern Israeli city of Ashkelon after warning people to leave by 17:00 local time (15:00 BST).
Air raid sirens have been sounding across Israel throughout the day, with the military saying more than 4,500 rockets have been fired since Saturday.
Israel says it’s killed two top Hamas officials while continuing to bomb the Gaza Strip; the group itself has not confirmed this.
A BBC reporter in Gaza said it was the worst bombing he had seen in 20 years, with neighbourhoods flattened.
Hamas has said it won’t negotiate on the hostages it has taken from Israel until the “end of the battle”.
Earlier, Israel said the border with Gaza was secure, three days after the Hamas attack that has killed 1,000 people on the Israeli side.
Meanwhile, more than 800 people have been killed by the strikes on the Palestinian side, according to the Hamas-run health ministry in Gaza.
The international community has called for a ceasefire, but both sides have so far refused to comply.