Gaza militants fired more than 250 rockets into Israeli towns and villages throughout the day on Saturday, while Israel hit back with tank shelling and airstrikes that Palestinian officials said killed four people.

Cross-border hostilities which broke out on Friday flared into a second day, with Palestinians fleeing Israeli strikes in Gaza and air-raid sirens sending Israelis running to shelters as interceptor missiles blew up rockets in the sky.

The escalation, which comes just ahead of both the Muslim holy month of Ramadan and Israel's Independence Day holiday,prompted Israeli Prime Minister Benjamin Netanyahu to convene his security chiefs.

Israel said one woman was seriously injured in a rocket strike on the Israeli city of Kiryat Gat, some 20 kilometres from the Gaza border, police said.

Police said a man was also hospitalised in the city of Ashkelon and spoke of other injuries without providing details.

Medics said the woman was 80 and the man was 50.

A house near Ashkelon was damaged while other rockets hit open areas.

The Israeli army said its tanks and planes hit some 120 militant targets in its response.

They included an Islamic Jihad attack tunnel that stretched from southern Gaza into Israeli territory, military spokesman Jonathan Conricus said.

Two multi-storey buildings in Gaza City were destroyed, residents said.

Israel said one of the buildings included Hamas military intelligence and security offices.

Turkey said an office for its state news agency Anadolu was located in the building and strongly condemned the strike.

The Gaza health ministry reported that a 14-month-old baby and her pregnant mother were killed in addition to two Palestinian men, while 40 were wounded.

It was not immediately clear if the two men were affiliated with militant groups.

An Israeli army spokeswoman said the military did not have any information on the incident involving the baby. The army said earlier it was targeting only military sites.

As the exchange of fire continued, Israeli Prime Minister Benjamin Netanyahu held consultations with security chiefs.

A statement from Hamas ally Islamic Jihad claimed responsibility for at least some of the rocket fire and said it was prepared for more if necessary.

Its armed wing distributed a video showing militants handling rockets and threatening key Israeli sites, including Ben-Gurion international airport near Tel Aviv.

Israel said it was closing its people and goods crossings with Gaza as well as the zone it allows for fishermen off the enclave until further notice due to the rocket fire.

Egyptian and UN officials were engaged in discussions to calm the situation, as they have done repeatedly in the past, while the European Union called for an immediate halt to rocket fire from Gaza.

The UN envoy for the Israeli-Palestinian conflict, Nickolay Mladenov, called on "all parties to immediately de-escalate and return to the understandings of the past few months."

The escalation follows the most violent clashes along the Gaza border in weeks on Friday.

Four Palestinians, including two Hamas militants, were killed after two Israeli soldiers were wounded in a shooting during weekly protests on the border.

Israel blamed Islamic Jihad for what it called the sniper attack, but stressed it held Hamas responsible for all violence from Gaza.

A ceasefire between Israel and Hamas brokered by Egypt and the United Nations had led to relative calm around Israel's 9 April general election.

But the past week had seen a gradual uptick in violence.

With the ceasefire at risk, a Hamas delegation led by its Gaza head Yahya Sinwar went to Cairo on Thursday for talks with Egyptian officials.

The ceasefire has seen Israel allow Qatar to provide millions of dollars in aid to Gaza to pay salaries and to finance fuel purchases to ease a severe electricity shortage.

Several factors may lead Israel to seek to calm the situation quickly.

Prime Minister Netanyahu is engaged in tough negotiations to form a new government following last month's election, while Israel is due to host the Eurovision song contest in Tel Aviv from 14-18 May.

The country also celebrates its Independence Day on Thursday.

In Gaza, the Muslim holy month of Ramadan begins in the week ahead.