Israel has bombed what it says are underground tunnels used by Hamas and Palestinian militants have fired rockets at Israeli cities as fighting spills into a second week.

After a night of Israeli air strikes on areas across the Hamas-run enclave, Israel's military said Gaza militants fired about 60 rockets towards Israeli cities overnight, down from 120 and 200 the two previous nights.

A Palestinian sponge factory in northern Gaza was hit in a Monday morning air strike and firefighters battled to quell the blaze, which sent plumes of smoke into the air.

One Palestinian was killed in an air strike later in the morning, medics said.

After rockets were fired from Gaza at the Israeli cities of Beersheba and Ashkelon, Israeli jets bombed what the military said were 15km of underground tunnels used by Hamas.

It also struck nine residences belonging to high-ranking Hamas commanders, the military said.

With the sounds of Israeli bombardment continuing throughout the morning, Gaza residents rushed to bakeries and drug stores to stock up on essentials.

"My children couldn't sleep all night even after the wave of intensive bombing stopped," said Umm Naeem, 50, a mother of five, as she shopped for bread in Gaza City.

Gaza health officials put the death toll since the hostilities flared at 198, including 58 children and 34 women. Ten people have been killed in Israel, including two children, Israeli authorities say.

Hamas began its rocket assault last Monday after weeks of tensions over a court case to evict several Palestinian families in East Jerusalem, and in retaliation for Israeli police clashes with Palestinians near the city's al-Aqsa Mosque, Islam's third holiest site, during the Muslim holy month of Ramadan.

World concern had already deepened after an Israeli air strike in Gaza that destroyed several homes on Sunday and which Palestinian health officials said killed 42 people, including 10 children, and persistent rocket attacks on Israeli towns.

Diplomatic channels working to end violence

US Secretary of State Antony Blinken on Monday urged all parties in the conflict to protect civilians and said the United States is working "intensively" to bring an end to the violence.

"We have been working around the clock through diplomatic channels to try to bring an end to the conflict," Mr Blinken said at a joint briefing with Denmark's foreign minister in Copenhagen.

Mr Blinken added that he has not seen any Israeli evidence of Hamas operating in a Gaza office building hit by an air strike over the weekend and has asked Israel for justification of the strike.

On Saturday, Israel bombed the 12-storey al-Jalaa Building, which housed the offices of The Associated Press, Al Jazeera and other media outlets, and claimed that Hamas used the building for a military intelligence office.

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At a UN Security Council meeting on Sunday, the US said it had made clear to Israel, the Palestinians and others that it was ready to offer support "should the parties seek a ceasefire".

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Prime Minister Benjamin Netanyahu said Israel's campaign in Gaza was continuing at "full force", and that deterrence had to be achieved to prevent future conflict with Hamas.

"We are acting now, for as long as necessary, to restore calm and quiet to you, Israel's citizens. It will take time," Mr Netanyahu said in a televised address after his security cabinet met on Sunday.

US President Joe Biden said his administration was working with all parties towards achieving a "sustained calm".

In New York, UN Secretary-General Antonio Guterres told the Security Council that the United Nations was "actively engaging all sides toward an immediate ceasefire" and urged them "to allow mediation efforts to intensify and succeed".

UN envoys have helped to mediate past truces between Israel and Hamas.

AP/Reuters