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Israeli Prime Minister Benjamin Netanyahu has vowed to press ahead with a fierce military offensive in the Gaza Strip, pushing back against calls from US President Joe Biden to wind down the operation that has left hundreds dead.

Netanyahu's tough comments marked the first public rift between the two close allies since the fighting began last week and could complicate international efforts to reach a cease-fire.

After a visit to military headquarters on Wednesday (Thursday morning AEST), Netanyahu said he "greatly appreciates the support of the American president", but said Israel will push ahead "to return the calm and security to you, citizens of Israel".

READ MORE: Strike from Gaza kills two as Israel topples six-storey building

He said he is "determined to continue this operation until its aim is met".

He spoke shortly after Biden told Netanyahu "that he expected a significant de-escalation today on the path to a ceasefire", the White House said.

Biden had previously avoided pressing Israel more directly and publicly for a cease-fire with Gaza's Hamas militant rulers. But pressure has been ramping up on him to intervene more forcefully as other diplomatic efforts also gather strength.

Netanyahu has repeatedly vowed to press ahead with the operation, and his tough response to Biden signalled he had no intentions of stopping. In another sign of potential escalation, militants in Lebanon fired a barrage of rockets into northern Israel on Wednesday.

READ MORE: Strike from Gaza kills two as Israel topples six-storey building

Egyptian negotiators have also been working to halt the fighting, and an Egyptian diplomat said that some of the country's top officials are waiting for Israel's response to a cease-fire offer. The diplomat spoke on condition of anonymity in line with regulations. Meanwhile, German Foreign Minister Heiko Maas said he hoped to fly to Israel for talks Thursday with Israelis and Palestinians.

Hundreds killed, including 64 children

Earlier in the day, the Israeli military said it was widening its strikes on militant targets in southern Gaza to blunt continuing rocket fire from Hamas. At least nine people were killed in the Gaza Strip on Wednesday.

The current round of fighting between Israel and Hamas began May 10 when the militant group fired long-range rockets toward Jerusalem after days of clashes between Palestinian protesters and Israeli police at the Al-Aqsa Mosque compound, a flashpoint site sacred to Jews and Muslims. Heavy-handed police tactics at the compound and the threatened eviction of dozens of Palestinian families by Jewish settlers had inflamed tensions.

READ MORE: Biden expresses 'support' for Israel-Gaza ceasefire

Since then, Israel has pounded Gaza with hundreds of airstrikes it says are targeting Hamas' militant infrastructure, and Hamas and other militant groups embedded in residential areas have fired more than 3700 rockets at Israeli cities, with hundreds falling short and most of the rest intercepted.

At least 227 Palestinians have been killed, including 64 children and 38 women, with 1620 people wounded, according to the Gaza Health Ministry, which does not break the numbers down into fighters and civilians. Hamas and Islamic Jihad say at least 20 of their fighters have been killed, while Israel says the number is at least 130. Some 58,000 Palestinians have fled their homes.

Twelve people in Israel, including a 5-year-old boy, a 16-year-old girl and a soldier, have been killed.

On Wednesday, militants in Lebanon fired a barrage of rockets into northern Israel, threatening to open up a new front in fighting.

The rocket attack, which drew Israeli artillery fire in response but apparently did not cause any injuries, raised the possibility of dragging Israel into renewed conflict with the powerful Lebanese militant group Hezbollah to its north.

READ MORE: Israel strikes Gaza tunnels as truce efforts remain elusive

No one immediately claimed responsibility for the attack, and Hezbollah, which fought a monthlong war against Israel in 2006, has stayed out of the fighting for now.

But the barrage appears to be carefully calibrated to send a political message that the group, which has tens of thousands of missiles, could join the battle at any time. Israel considers Hezbollah to be its most formidable threat, and has threatened widespread destruction in Lebanon if war were to erupt.

The rockets are widely believed to be fired by Palestinian factions based in south Lebanon. But they cannot operate without Hezbollah's tacit consent.

'We ask God to help us'

In the Gaza Strip, one of the Israeli airstrikes destroyed the home of an extended family.

Residents surveyed the piles of bricks, concrete and other debris that had once been the home of 40 members of al-Astal family in the southern Gaza town of Khan Younis. They said a warning missile struck the building five minutes before the airstrike, allowing everyone to escape.

READ MORE: Sydney charity worker tells of conflict's 'heartbreaking' toll on children

Ahmed al-Astal, a university professor, described a scene of panic, with men, women and children racing out of the building. Some of the women didn't even have time to cover their hair with Muslim headscarves, he said.

"We had just gotten down to the street, breathless, when the devastating bombardment came," he said. "They left nothing but destruction, the children's cries filling the street. … This is happening, and there is no one to help us. We ask God to help us."

The Israeli military said it struck a militant tunnel network around the towns of Khan Younis and Rafah, with 52 aircraft hitting 40 underground targets.

Among the nine people killed Wednesday was a reporter for Hamas-run Al-Aqsa radio and two people who died when warning missiles crashed into their apartment.

Military officials, meanwhile, said a mysterious explosion that killed eight members of a Palestinian family on the first day of the fighting was caused by a misfired rocket from Gaza. "This wasn't an Israeli attack," said Lt. Col. Jonathan Conricus, a military spokesman.

Medical supplies, water running low

Since the fighting began, Gaza's infrastructure, already weakened by a 14-year blockade, has rapidly deteriorated. Medical supplies, water and fuel for electricity are running low in the territory, on which Israel and Egypt imposed the blockade after the Islamic militant group Hamas seized power in 2007.

Israeli attacks have damaged at least 18 hospitals and clinics and destroyed one health facility, the World Health Organisation said. Nearly half of all essential drugs have run out.

The Gaza Health Ministry said it had salvaged coronavirus vaccines after shrapnel from an Israeli airstrike damaged the territory's only testing facility, which also administered hundreds of vaccines. The operations were relocated to another clinic.

Dr Majdi Dhair, head of preventive medicine at the ministry, said the territory was already struggling to recover from a coronavirus wave that hit in February, with more than 4200 active cases. At least 986 people have died from COVID-19 in Gaza, which only has enough supplies to vaccinate some 55,000 people out of a population of 2 million.

Among the buildings levelled by Israeli airstrikes was one housing The Associated Press' Gaza office and those of other media outlets.

Netanyahu alleged that Hamas military intelligence was operating in the building. Secretary of State Antony Blinken said Tuesday that Israel had given the US information about the bombing, without elaborating.

The AP has called for an independent investigation. The news organisation's president, Gary Pruitt, has said the AP had no indication Hamas was present in the building and that "this is something we check as best we can."

The fighting, the worst since a 2014 war between Israel and Hamas, has ignited protests around the world and inspired Palestinians in Israel and the occupied territories to call a general strike Tuesday. It was a rare collective action that spanned boundaries central to decades of failed peace efforts. Israel captured the West Bank, east Jerusalem and Gaza in the 1967 Mideast war, territories the Palestinians want for their future state.

When baby Malakai Paraone died in 2016, failed by Western Australia's health system, promises were made to his family.

His grandmother, Julia Brown, wants to know why neither Health Minister Roger Cook nor the previous government introduced Malakai's Rule, a system that was proposed to give worried parents more chance to be heard in the children's hospital system.

"We were promised a lot of things, and it never came through," Ms Brown told 9News on a video call from a remote part of New Zealand.

READ MORE: Coronial inquest into Perth baby's death after hospital visits

READ MORE: Doctor and nurses reported to regulator over Aishwarya's hospital death

A similar system will now be brought in because another child, seven-year-old Aishwarya Aswath, died in similar harrowing circumstances on Easter Saturday this year.

"It brought back raw memories. It brought back memories of Malakai. So yeah, I feel for the family," Ms Brown said.

"I know what they would have been going through."

The tragedy of Malakai's death unfolded across five days in 2016. On Monday, August 22, the seven-month-old became unwell and his parents took him to Midland Hospital. They went home after being told he was teething.

On Tuesday an ambulance took him to Princess Margaret Hospital after a high heart rate and vomiting but he was sent home with Panadol. 

READ MORE: Aishwarya's parents demand hospital reform

The next day, a GP diagnosed a virus and on the Thursday the parents rushed back to PMH.

Within hours their baby was in intensive care. His life support was turned off the next day.

"We pushed hard, and we went through a traumatic time trying to push for it and dealing with other things," Ms Brown said.

Mr Cook is now promising a new rule called Aishwarya's Care.

READ MORE: Schoolgirl 'should have had more help'

On Wednesday, 9News pushed the minister on why he didn't deliver Malakai's Rule when the Labor Party came to power in 2017.

"I raised this issue with the department and I asked them if we should have a similar system, they actually said that they had a system in place called Care Call," he said.

"Care Call obviously did not do its job on the third of April (when Aishwarya died)."

But Care Call was never available to parents in the emergency department.

Premier Mark McGowan admitted Malakai's Rule would have improved on the system already in place.

Malakai's Rule would have ensured a failsafe for parents: a phone number they could call if feeling ignored in emergency, like Aishwarya's parents were for an hour and 40 minutes before her death.

READ MORE: Aishwarya's parents demand hospital reform

Her death is history repeating for Malakai's family, who received an apology at the time from former health minister John Day and outrage from Mr Cook, who was then in opposition.

"I can't understand how this family has been let down so badly," Mr Cook said at the time, even writing to the Australian Medical Association about getting the new system implemented quickly.