Israel rejected a ceasefire offer from Hamas on Monday night, vowing to continue with its invasion of the southern Gazan city of Rafah and saying a last-minute proposal had failed to meet its key demands.

Hamas officials said on Monday that they had agreed to a ceasefire deal mediated by Egypt and Qatar that would see it release hostages in exchange for the end of the Israeli bombardment of Gaza.

But Israel said the terms of the deal were “far from” its “obligatory demands” and that it would continue with a planned ground invasion of Rafah “in order to apply military pressure to Hamas” and to free hostages.

Benjamin Netanyahu’s office said it would send a negotiating team to Egypt to discuss an agreement, but officials earlier warned that the Hamas plan “would appear to be a ruse intended to make Israel look like the side refusing a deal”.

On Monday night, Israeli jets launched a bombardment of Rafah, striking 50 targets according to the IDF. Earlier in the day, Israel signalled that it had begun its long-awaited offensive on the city, warning 100,000 people to evacuate an eastern section.

The launch of the operation came despite fierce opposition from Israel’s key allies. Joe Biden, the US president, urged Mr Netanyahu not to go ahead with the invasion in the hours before news broke of Hamas’s ceasefire offer.

“We’ve made clear our views about operations in Rafah that could potentially put more than a million innocent people at greater risk,” said John Kirby, the White House’s national security spokesman.

“During this call with prime minister Netanyahu, the president again made this clear. He also made clear that we continue to believe that the hostage deal is the best way to avoid that sort of an outcome.”

Mr Kirby added that the White House was examining Hamas’s ceasefire proposal and would soon communicate its views to Qatar, Israel and Egypt. He said Mr Netanyahu had agreed to keep aid flowing into Gaza through Kerem Shalom, the key crossing into the south of the Strip.

Four Israeli soldiers were killed in an attack at the crossing on Sunday, which was claimed by the armed wing of Hamas.

On Monday, Rishi Sunak said he was “deeply concerned” by the prospect of a Rafah offensive, and Sir Keir Starmer, the Labour leader, called for it to be dropped entirely.

Jordan’s King Abdullah, who met Mr Biden in Washington on Monday, said the Israeli attack on Rafah threatened a “new massacre”.

Palestinians had taken to the streets of the city on Monday to celebrate news of a potential peace deal – but the launch of Israeli strikes dampend hopes that Rafah would be spared.

“There was a lot of tension in the morning because of the leaflets the army was dropping [warning people to evacuate] – many people were distressed and tried to leave, roads were clogged,” Said, a 35-year-old, told The Telegraph on Monday night.

“In the evening, we got the news. People want to be happy, but they’re cautious and waiting, and we still hear bombings. We really hope the war in Gaza will stop and we will see better days.”

Khalil al-Hayya, a senior Hamas member, said the group had signed up to an Israeli-backed deal in three stages of 42 days, with hostages released in the first stage and the war ended in the second stage.

He said the deal would also involve the return of displaced Palestinians to their homes in Gaza, and the release of Palestinian prisoners held by Israel.

Israeli sources told local media that the terms suggested by Hamas on Monday had been “softened” since a framework agreement proposed by Israel last week, which would have involved three prisoners being released every three days the ceasefire lasted.

The deal supported by Hamas would instead have seen three hostages released each week, according to Israeli media reports.

Foreign diplomats briefed on the ceasefire negotiations later told Reuters that the deal Hamas offered was close to that proposed by Israel on April 27. “What happens next depends on Israel to accept or reject Hamas’s response. The ball is in Israel’s court,” said a source.

One US official told Reuters that Mr Netanyahu had “not appeared to approach the latest phase of negotiations [with Hamas] in good faith”.

Rear Admiral Daniel Hagari, a spokesman for the Israel Defense Forces (IDF), said Israel was “exploring every single thing that we hear”, but that no deal had been reached.

“We are exhausting the potential of negotiations and bringing back the hostages and that is our main mission, to bring them home as quickly as possible,” he added. “But in parallel, we are continuing to act in an operational manner in the Gaza Strip, and we will continue to do so.”

The months-long negotiations over a ceasefire have been held up by disagreements over the number of hostages to be released by Hamas, and whether the war would end once they had been returned.

Despite pressure from the US, Mr Netanyahu has said he will continue with the Rafah offensive to “eliminate Hamas’s battalions there, with or without a deal”.

Hamas’s acceptance of a ceasefire deal increased internal pressure on Mr Netanyahu to end the war, with the Israeli Hostages and Missing Families Forum welcoming the Hamas announcement and calling for an agreement.

The group said: “Now it’s time for the Israeli government to prove, with action, its commitment to its citizens. The cabinet must take Hamas’s acceptance and turn it into a deal for returning all [hostages].”

But Itamar Ben-Gvir, the national security minister, said Hamas was playing “games” and called for Israel to occupy Rafah immediately, adding: Military pressure must be increased until the complete defeat of Hamas and its absolute surrender.”