Around 200 British nationals are trapped in the besieged enclave, the prime minister said, as Israel continues to step up its preparations for a full-scale ground invasion.
Mr Sunak said he wanted to bring those caught up in the fighting home and promised that, while the UK cannot do so “immediately ... when the moment arises – we’ll be ready to take it quickly”.
His pledge came as Israel unleashed its heaviest bombardment of Gaza, targeting tunnel shafts and rocket launchers used by Hamas militants, with the country’s military saying it had launched 250 airstrikes in the past 24 hours.
It insisted the push would prepare “for the next stages of the war” nearly three weeks after Hamas terrorists murdered more than 1400 people in Israel.
It comes as:
Israeli tanks crossed into the besieged enclave overnight
Prime minister Benjamin Netanyahu vowed a widely anticipated full ground offensive would go ahead but gave no timeline
The United Nations Agency for Palestinian Refugees (UNRWA) said it urgently needed fuel to maintain life-saving operations
The UK hopes Britons will be able to leave via the Rafah crossing into Egypt, through which some aid has been transported into Gaza
Mr Sunak said he had discussed the plight of the trapped Britons with the leaders of both Egypt and Israel and said he was “very keen to be able to bring them out – and bring them home”.
He is pushing for a temporary “pause” in the fighting – as distinct from a ceasefire – to allow aid into the Gaza strip, amid growing fears of a full-scale humanitarian crisis.
Mr Sunak also expressed hopes this would create a “safer environment” that could allow UK citizens, and potentially hostages held by Hamas, to leave the bombarded 25-mile strip.
At the weekend, the first aid convoy made it through the crossing into Gaza. But Cairo has reportedly blamed Israeli bombings around Rafah for the reason it cannot open the crossing for foreign nationals to pass through.
No 10 is understood to not expect any breakthrough at the crossing imminently but Border Force teams have been poised in Egypt for some time, ready to evacuate those trapped as soon as the border opens.
Ministers are keen not to see a surge of people down towards a border they cannot cross, where they could end up in a worse position. Ministers have been encouraging British nationals and those with dual nationality to register with the Foreign Office for help getting out.
The 200 people who have made contact so far are understood to make up most of those who are keen to leave when the border opens. It is thought they could leave Egypt for Britain on commercial flights, which are still operating and can often be faster than chartering planes.
The prime minister said the government recognised that for British nationals to leave Gaza there “needs to be a safer environment, which of course necessitates specific pauses, which are distinct from a ceasefire”.
He added: “We’re very keen to be able to bring them out and bring them home. What I can tell you is we've pre-positioned Border Force teams to Egypt.
“So that if there is a possibility for our nationals to cross the Rafah crossing, we’re ready to get them in and bring them back. It is not something we can do immediately but when the moment arises, we'll be ready to take it quickly.”
It comes as the Israeli military said that troops and tanks launched a brief ground raid into northern Gaza overnight into Thursday, in order to “prepare the battlefield” ahead of a widely-expected ground invasion.
Downing Street said it would not speculate about whether it would support a ceasefire if Israel succeeded in rescuing hostages from Gaza.
“I’m not going to speculate on what we may do in the future. This is a fluid situation. I’ve set out our position currently,” the prime minister’s official spokesperson said.
The health ministry in the Hamas-run strip has said that more than 7,000 Palestinians have been killed in the war – a figure that includes the disputed toll from an explosion at a hospital.
That is more than three times the number of Palestinians killed in the six-week-long Gaza war in 2014, according to UN figures.
It said the latest strikes killed 750 people, the highest reported death toll in a 24-hour period so far.
Israel has called for Palestinians to evacuate from northern to southern Gaza, but UN officials have said that “nowhere is safe” in the strip.
The foreign ministers of the United Arab Emirates, Jordan, Bahrain, Saudi Arabia, Oman, Qatar, Kuwait, Egypt and Morocco all condemned the targeting of civilians and violations of international law in Gaza, in a joint statement on Thursday. It mentioned that the right to self-defence does not justify breaking law and neglecting Palestinians’ rights. The Arab foreign ministers also condemned forced displacement and collective punishment in Gaza.
Meanwhile, Turkish President Recep Tayyip Erdogan told Pope Francis that the international community's silence over what was happening in Gaza was a “shame for humanity”. It comes as EU leaders are set to call for the establishment of “humanitarian corridors and pauses” to get urgently needed aid into Gaza as fuel supplies are days from running out.