Israel says Palestinian militants in Gaza have fired more than 460 rockets at it, and that its military has hit over 130 militant targets in Gaza, in the heaviest fighting in nine months.
Six people were killed and 45 injured in Gaza, local medics say.
Several were hurt rushing to shelters in Israel, where most rockets have been intercepted or fell in open areas.
It comes a day after 15 Palestinians were killed in Israeli strikes on Gaza, including three Islamic Jihad leaders.
The Islamic Jihad (PIJ), which is the second biggest militant group in the territory after Hamas, had sworn to avenge their deaths.
In a televised address on Wednesday night, Israeli Prime Minister Benjamin Netanyahu warned that Israel’s “campaign is not over”.
“We have hit Islamic Jihad with the most significant blow it has ever suffered,” he said, referring to the simultaneous killings of the three PIJ commanders in the early hours of Tuesday.
Hours before Mr Netanyahu spoke, Egyptian media reported that Egypt had brokered a ceasefire, but there was no immediate confirmation from the two sides. Soon afterwards, another rocket barrage was fired towards southern Israel and there were further strikes in Gaza.
An umbrella organisation representing armed factions in Gaza earlier warned that “if Israel increases its aggression, dark days await it”.
Palestinians said the exchange of fire began on Wednesday morning with several loud explosions in southern Gaza, sending up large plumes of smoke.
The Israel Defense Forces (IDF) said an aircraft targeted PIJ operatives travelling in a vehicle to a concealed rocket launcher in the Khan Younis area.
About an hour later, the IDF announced that it had started attacking underground rocket launchers belonging to PIJ across the territory in order to thwart planned attacks.
Palestinian media reported strikes in and around Gaza City, in the southern town of Rafah, and in the northern towns of Beit Lahiya and Beit Hanoun.
The Palestinian health ministry in Gaza reported that six people were killed in Israeli strikes.Four of them were members of the military wing of the Popular Front for the Liberation of Palestine, the group said, adding that two died in Khan Younis and the other two in Rafah.A 10-year-old girl was also killed in Gaza City in unclear circumstances.
As Israeli aircraft struck Gaza, militants began firing barrages of rockets from Gaza and rocket sirens sounded in communities in southern Israel.
Later, rockets were launched towards central Israel, triggering sirens in the Tel Aviv area, 80km (50 miles) from Gaza. In one video filmed in Old Jaffa, explosions could be heard as two rockets appeared to be intercepted overhead.
The Israeli military said four houses suffered direct hits – two in Sderot, which is only 1km (0.6 miles) from Gaza, and two in Ashkelon, 7km from Gaza. Another hit the roof of a kindergarten in Nirim, on the edge of Gaza to the south, and a yeshiva (religious school) in Netivot, 11km east of Gaza. There were no injuries directly from rocket fire.
The IDF said one in four rockets fired at it had fallen short and landed inside Gaza. It said Israel’s Iron Dome air defence system had intercepted 153 rockets, three had hit urban areas in Israel and the rest landed in open areas.
The Joint Operations Room of armed groups in Gaza, which includes Islamic Jihad and Hamas, claimed in a statement that they had launched the rockets.
“The damage to the homes of civilians and faction fighters is a red line, and we will respond strongly to it. Resistance forces are ready for all options,” it said.
The IDF launched Operation Shield and Arrow in the early hours of Tuesday with several waves of strikes across Gaza that killed 13 Palestinians.
Three were PIJ commanders who the IDF said were involved in recent attacks against Israeli civilians and were planning more. But the other 10 dead were civilians, including four women and four children.
Another two Palestinians were killed on Tuesday afternoon in a strike that the IDF said targeted militants planning to fire anti-tank missiles.
The strikes were the deadliest since three days of hostilities between Israel and PIJ last August, in which 49 Palestinians were killed in Gaza.
PIJ has been responsible for many of the rocket attacks on Israel in recent years and is sworn to Israel’s destruction.
There was a serious flare-up last week, as PIJ and other groups fired more than 100 rockets into Israel over two days, following the death in an Israeli prison of a Palestinian hunger striker. The Israeli military carried out air strikes on sites it said were linked to Hamas in response.
Tensions also remained high in the occupied West Bank on Wednesday, after Israel carried out arrest raids overnight.
Two Palestinians were killed in the town of Qabatiya by Israeli forces, who said the pair fired at them. The IDF also said a soldier was also seriously wounded during a separate exchange of fire with Palestinian gunmen in Tubas.