Witnesses and medics said Israeli jets carried out at least five air strikes late on Monday in the northern town of Beit Lahiya and Gaza City.
Witnesses said a security compound for Hamas, which controls the Gaza Strip, a training camp north of the city and a brickworks and metal foundry in northern Gaza were among the targets.
The Israeli military confirmed one of the strikes, saying it was targeting several Hamas-affiliated fighters in
northern Gaza, as well as a tunnel that it said was used to smuggle weapons.
The military said the strikes were in response to Palestinian rocket fire into Israel.
The Ezzedine al-Qassam Brigades, the military wing of Hamas, had earlier said it would stop cross-border fire into Israel if the Israelis halted attacks on Gaza.
It said rocket fire into Israel on Saturday had been in response to an Israeli strike last week which killed two of its members, but the group said it was ready to call an end to the tit-for-tat violence if Israel also did so.
"If the enemy stops the escalation and aggression against our people we will implement the Palestinian national agreement," the AFP news agency reported, citing a statement from the group.
The statement referenced a truce reaffirmed by Gaza's main armed factions in January.
The offer, however, came with a warning attached: "The enemy will pay a heavy price if it continues its aggression and crimes against our people in the Gaza Strip."
In a later statement, Hamas spokesman Taher al-Nunu said the movement's Gaza government was committed to preserving the informal truce, with the backing of other groups, AFP reported.
"The government affirms that there is consensus among the factions regarding the security situation in the Strip," he said.
Also on Monday, Danny Ayalon, Israel's deputy foreign minister, issued a threat against Hamas leaders.
"If Hamas decides to escalate, we will put an end to it... We have several actions before putting ground forces in Gaza, including direct threats against Hamas leaders," Ayalon told public radio.