The strike on the building in Gaza City wounded 18 Palestinians.
Gaza militants had at around noon halted fire after targeting Israel with some 180 rockets and mortars beginning Wednesday night and into Thursday.
The former prime minister says that he has previously explored ways to remove Hamas from power in Gaza, but his proposal, which apparently included an Israeli reconquest of the Strip, was rejected by the PA and Arab states.
Beginning on Wednesday night and into Thursday, fireballs and explosions shook the Gaza Strip while plumes of smoke rose from the enclave.
Hamas warned Israel it would pay for the attack, and on Wednesday the army said shots were fired at civilian engineers working on an underground barrier along the border, prompting the military to respond with tank fire.
The Said Meshal Cultural Centre confirmed on its Facebook page the offices in Gaza City had been destroyed, saying the building also hosted offices for the Egyptian community in Gaza.
The health ministry added that around 12 Palestinians were injured, two critically, and were transferred to the hospital for treatment.
Earlier in the day, Hamas deputy chief in Gaza Khalil al-Hayya said efforts to reach a truce with the Israeli regime were at "an advanced stage".
"Factions of the resistance consider this round of escalation over as far as we are concerned, and the continuation of calm depends on the behaviour of the occupation", the Palestinian official said, using militant groups' term for Israel.
According to medical officials, the Israeli regime's airstrikes killed Enas Khammash, a pregnant Gazan woman, and her 18-month-old daughter Bayan, as well as another Palestinian.More news: Kawhi Leonard pens letter thanking Spurs, Gregg Popovich
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Israel's military declined to comment.
Abu Mujahed, spokesman for the Popular Resistance Committee, a smaller faction in Gaza, said on his Facebook page: "Upon Egyptian and global efforts, a ceasefire between Palestinian resistance began conditional on the commitment of the occupation".
A Hamas spokesperson, Abdul Latif al-Qanoa, said that Hamas militants were "defending its people against the ongoing Israeli aggression on Gaza as part of its national duty". "In response, the IDF targeted over 150 terror targets in the Gaza Strip", the IDF said on Twitter.
The escalation, one of the most intense exchanges of fire since the 2014 war between Israel and Hamas, came amid Egyptian-brokered talks in Cairo over a cease-fire between the two sides.
"What I can say is that we targeted by definition only military targets that were clearly used by Hamas where there was an active presence of Hamas and nothing else", he told journalists on Thursday morning. One Israeli soldier was killed by a Gaza sniper during this period. Some rockets landed in the communities, and many others were shot down by the Iron Dome, which is created to intercept only rockets headed towards population centers.
Palestinian president Mahmud Abbas called on the global community to "immediately and urgently intervene".
UN Middle East envoy Nickolay Mladenov said the United Nations had engaged with Egypt in an "unprecedented effort" to avoid serious conflict, but cautioned that "the situation can rapidly deteriorate with devastating consequences for all people".
Gazans have held weekly protests since March 30, some of which have become violent, to demand the right of return for Palestinian refugees.
At least 165 Palestinians have killed by Israeli fire since then.