He said that the raids were in response to a wave of 56 rocket and mortar attacks from Gaza into Israel since Saturday.
Another tunnel was hit in a single and separate attack early Monday evening, the military said.
Palestinian witnesses said that among targets in the north and centre of the strip were a police post of Hamas, which rules Gaza, and a training facility of its military wing, the Ezzedine al-Qassam Brigades which earlier offered to stop cross-border fire into Israel if the Israelis halted attacks on Gaza.
A factory producing construction materials and a metal workshop were also hit, while in southern Gaza there were at least three strikes in the vicinity of Khan Younis town, witnesses said.
One targeted a group of fleeing activists, one a building belonging to the town council and another an empty plot, they said, adding that no casualties were recorded.
Gaza residents reported receiving telephone calls from the Israeli military urging them to leave the vicinity of Hamas facilities.
Ezzedine al-Qassam, which lobbed about 50 mortar rounds into Israel on Saturday, made its truce offer in a statement released after Israeli aircraft made their first raid on the enclave on Monday evening.
It said Saturday's barrage had been in response to an Israeli strike last week which killed two of its members, but that 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 statement said, referring to a truce reaffirmed by the main militant factions in January.
The offer, however, came with a warning attached:
"The enemy will pay a heavy price if it continues its agression and crimes against our people in the Gaza Strip," the statement added.
Israeli military and government officials declined to comment, but the Jewish state's often-stated standing policy is to "respond forcefully" to every Palestinian attack.
Witnesses said the target of the first raid was a car repair workshop east of Gaza City, owned by the powerful Doghmush clan which has links to Islamic militants.
Also on Monday, Israeli Deputy Foreign Minister Danny Ayalon issued a death 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.
A rocket fired from Gaza overnight on Sunday exploded in southern Israel, causing neither casualties nor damage, several hours after another rocket exploded harmlessly in the town of Ashkelon.
After Saturday's mortar fire Israel pounded Gaza, wounding at least five Palestinians and cutting power supplies.
The mortar attacks, the fiercest since Israel carried out a 22-day offensive codenamed "Operation Cast Lead" against Gaza rocket fire in December 2008 and January 2009, wounded two Israelis and caused minor damage.
In January this year, Gaza's main militant factions confirmed a year-old truce after weeks of increased rocket fire and spiralling tensions along the border prompted a warning from Arab leaders that Gaza risked a major new Israeli invasion.
On Saturday Israeli opposition leader Tzipi Livni called for just that in response to the mortar barrage.
"The right way to deal with it is with force, just like Israel did during and after Operation Cast Lead," news website Ynet quoted her as telling local authority heads in the border region.