Witnesses reported extensive damage to the building, which Egypt’s PM Hisham Qandil had visited on Friday.
At least 38 Palestinians and three Israelis have died since Israel killed Hamas’s military chief on Wednesday.
Israel earlier put 75,000 reservists on stand-by amid speculation of an impending ground invasion.
Militants in Gaza have continued to fire rockets into Israel, after aiming at Tel Aviv and Jerusalem on Friday.
Following a lull, Gaza City was hit by a string of large explosions shortly after 03:00 local time (01:00 GMT) on Saturday.
There was another series of strikes in and around the city after 05:00 local time, with several targeting Hamas’ cabinet buildings, which correspondents say were likely to have been empty.
One of the targets was the house of a Hamas leader in Jabaliya, north of Gaza City.
The BBC’s Paul Danahar tweeted from the scene: “A mother in her wrecked home… is scurrying around collecting her daughters dolls, dusting them off.”
At least eight Palestinians are reported to have been killed in overnight strikes. The dead are said to include three members of the Izz al-Din al-Qassam Brigades, Hamas’s military wing.
Israel issued a statement saying it was targeting rocket launching squads and weapons storage facilities and smuggling tunnels on the border with Egypt in southern Gaza.
Israeli military spokeswoman Avital Leibovich said 200 targets had been hit overnight, including 120 rocket launchers.
There are rumours that a ground attack is imminent, but Israeli officials say no decision has been made.
Israel blocked access to three major routes leading into Gaza on Friday. Call-up papers have already been sent to 16,000 Israeli reservists, with officials authorising the mobilisation of another 75,000.
Militants and civilians, including at least seven children, have been among the Palestinians killed during Israeli strikes in recent days, Hamas says.
The group’s military leader Ahmed Jabari was killed on Wednesday. A senior commander was killed on Friday, officials said.
Two Israeli women and a man died when a rocket hit a building in the southern town of Kiryat Malachi on Thursday, officials said.
Before the recent offensive, Israel had repeatedly carried out air strikes on Gaza, as Palestinian militants fired rockets across the border.
On Friday, Hamas said it fired rockets at Tel Aviv and Jerusalem: Haaretz newspaper said it was the first time since 1970 that a rocket had been fired at Jerusalem.
Israel’s army says the operation – codenamed Pillar of Defence – has hit more than 800 sites in Gaza, including underground rocket launchers & infrastructure.
It says hundreds of rockets have been fired into Israel from Gaza since Wednesday, a quarter of which had been intercepted by its radar defence system, Iron Dome.
The army said about 10 rockets were fired from Gaza early on Saturday and that three soldiers were injured one rocket strike in southern Israel.
Palestinian Authority President Mahmoud Abbas has accused Israel of carrying out “massacres”.
Tunisian Foreign Minister Rafik Abdessalem has arrived in Gaza through the Rafah border crossing from Egypt to show support for Hamas.
Western leaders and UN Secretary General Ban Ki-moon have appealed for both sides to stop the violence.
In a phone call with Israeli Prime Minister Benjamin Netanyahu on Friday, President Barack Obama reiterated US support for Israel’s “right to defend itself”.
Mr Obama also spoke to Egyptian President Mohammed Mursi. Mr Mursi has called the Israeli raids “a blatant aggression against humanity” and promised that Egypt “will not leave Gaza on its own”.
Ties between Hamas and Egypt have strengthened since Mr Mursi’s election earlier this year.
Hamas was formed as an offshoot of the Muslim Brotherhood, to which Mr Mursi belongs.