There is a "narrow window" to prevent famine from spreading further in Gaza, a top UN official said on Sunday, calling on Israel to allow unimpeded aid delivery in the territory, where it is fighting Palestinian militant group Hamas.

According to a global hunger monitor, hundreds of thousands of Palestinians are already experiencing or at risk of famine in areas including Gaza City, the enclave's largest urban centre, where Israel has launched a new offensive against Hamas.

Israel, which stopped all aid for 11 weeks from March until mid-May, says it is doing more to let aid enter and be distributed in the enclave to prevent food shortages, though international agencies say far more is needed.

"There is a narrow window – until the end of September – to prevent famine from spreading to Deir al Balah (in central Gaza) and Khan Younis (southern Gaza). That window is now closing fast," said United Nations aid chief Tom Fletcher.

Cogat, the Israeli defence agency that deals with humanitarian issues, said on Sunday that over the past week aid from more than 1,900 trucks, most supplying food, was distributed in Gaza.

"We will continue facilitating humanitarian aid into Gaza for the civilian population - not Hamas," Cogat said in a statement.

Israel last month launched an assault on the outskirts of Gaza City and its forces are now just a few kilometres from the city centre, where it issued warnings over the weekend to civilians to evacuate high rise buildings it says are being used by Hamas, before bombing them.

Israel did not provide evidence to show Hamas was using the buildings, an accusation the militant group denied.

Overnight, strikes killed 14 people across the city, local health officials said, including a strike on a school in southern Gaza City sheltering displaced Palestinians.

The Israeli military said it had struck a Hamas militant and that civilians had been warned before the strike was carried out.

The military on Saturday also warned Gaza City's civilians to leave for the south, including Khan Younis, where hundreds of thousands of Palestinians are already sheltering in cramped tent encampments along the coast. (Reuters)