Israel and Hamas have continued to trade fire as the war between the rivals entered a second week.

Israel carried out a wave of air strikes on what it said were militant targets in Gaza, levelling a six-storey building in Gaza City, and Palestinian militants fired dozens of rockets into Israel.

Explosions from the air strikes echoed through the pre-dawn darkness in Gaza City, sending flashes of orange across the night sky.

The strikes toppled the Kahil building, which contains libraries and educational centres belonging to the Islamic University. Clouds of dust hung over the site, which had been reduced to piles of concrete rubble and tangled power lines.

There were no immediate reports of casualties from the overnight strikes.

Heavy fighting broke out on May 10 when Gaza's militant Hamas rulers fired long-range rockets towards Jerusalem in support of Palestinian protests in the city against Israel's heavy-handed policing of the Al-Aqsa mosque compound, a flashpoint holy site sacred to Jews and Muslims, and the threatened eviction of dozens of Palestinian families by Jewish settlers.

The Israeli military said on Tuesday it fired more than 100 munitions at 65 militant targets, including rocket launchers, a group of fighters and the homes of Hamas commanders that the army said were being used for military purposes. It said more than 60 fighter jets took part in the operation.

The military said Palestinian militants fired 90 rockets, 20 of which fell short into Gaza. Israel says its missile defences have a 90% interception rate.

At least 212 Palestinians have been killed in the week of heavy air strikes, including 61 children and 36 women, with more than 1,400 people wounded, according to the Gaza Health Ministry.

Ten people in Israel, including a five-year-old boy and a soldier, have been killed in the ongoing rocket attacks launched from civilian areas in Gaza towards civilian areas in Israel.

Israel says it has inflicted heavy damage on Hamas' military infrastructure, including a vast network of militant tunnels it refers to as the "Metro."

On Monday, US President Joe Biden expressed support for a ceasefire between Israel and Gaza's militant Hamas rulers in a call to Israeli Prime Minister Benjamin Netanyahu.

However, Mr Biden stopped short of demanding an immediate stop to the eight days of Israeli air strikes and Hamas rocket barrages.

Mr Biden's carefully worded statement came with the administration under pressure to respond more forcefully despite its determination to move the US foreign policy focus away from Middle East conflicts.

His comments on a ceasefire were open-ended, and similar to previous administration statements of support in principle for a ceasefire. That is in contrast to demands from dozens of Democratic politicians and others for an immediate halt by both sides.

The US leader "encouraged Israel to make every effort to ensure the protection of innocent civilians", the White House said in a statement about the call.

An administration official familiar with the call said the decision to express support and not explicitly demand a ceasefire was intentional.

While Mr Biden and top aides are concerned about the mounting bloodshed and loss of innocent life, the decision not to demand an immediate halt to hostilities reflects White House determination to support Israel's right to defend itself from Hamas, the official said.

Separately, the United States, Israel's top ally, blocked for a third time on Monday what would have been a unanimous statement by the 15-nation UN Security Council expressing "grave concern" over the intensifying Israeli-Palestinian conflict and the loss of civilian lives.

The final US rejection killed the Security Council statement, at least for now.

White House press secretary Jen Psaki and national security adviser Jake Sullivan said the United States was focusing instead on "quiet, intensive diplomacy".