"It was a combined ground raid which happened through paragliders, through the sea and through the ground," occupation army spokesman Richard Hecht told reporters.

"Right now we're fighting. We're fighting in certain locations around the Gaza Strip... our forces are now fighting on the ground" in Israel, he added.

Hecht confirmed there are casualties but would not be drawn on details, nor discuss reports that multiple Israelis had been captured by Palestinian militants.

Thousands of military reservists are set to be drawn up for Gaza, as well as northern Israel, near foes Lebanon and Syria, and the occupied West Bank.

"We're looking at all arenas... we understand this is something big," Hecht said.

At least 2,200 rockets were fired from Gaza by around 10:30 am (0730 GMT), the spokesman said, while Hamas militants put the figure at more than 5,000.

Israel imposed a crippling blockade of Gaza in 2007 after Hamas took control and there have been multiple cross-border wars since.