The Prime Minister started work on a plan with France and Germany over the weekend after an emergency call with the two countries’ leaders, Emmanuel Macron and Friedrich Merz.
His official spokesman had said earlier that he planned to share details with key allies, including Arab states, in the coming days.
In a readout of his meeting with the US president, Downing Street said the two reiterated calls for an immediate ceasefire to pave the way for peace.
“The Prime Minister welcomed the President’s efforts to secure this, and shared the plans he is working on with other European leaders to bring about a lasting peace,” a No 10 spokesperson said.
Sir Keir described the situation in Gaza as an “absolute catastrophe” as he appeared alongside Mr Trump at his Turnberry golf course ahead of their talks.
The British public is “revolted” at scenes of desperation in the Palestinian territory, he said.
As the pair faced the press, Mr Trump suggested the US would set up new food centres in Gaza without fences, after the current US-led arrangement had led to multiple killings of Palestinians.
“It’s a humanitarian crisis, it’s an absolute catastrophe,” the Prime Minister said of the situation in Gaza as he arrived in Ayrshire.
He added: “Nobody wants to see that. I think people in Britain are revolted at seeing what they’re seeing on their screens, so we’ve got to get to that ceasefire.”
Israel announced at the weekend that it would suspend fighting in three areas of Gaza for 10 hours a day and open secure routes for aid delivery, while the UK confirmed it was taking part in plans led by Jordan to airdrop aid into the territory.
Speaking at his Ayrshire golf course, Mr Trump said: “We’re going to set up food centres, and we’re going to do it in conjunction with some very good people.”
He added: “And other nations are joining us. I know your nation’s joining us, and we have all of the European nations joining us, and others also called and they want to be helpful.
“So we’re going to set up food centres where the people can walk in and no boundaries, we’re not going to have fences.”
Other nations are “going to have to step up” when it comes to helping people in Gaza, the US President added.
Hinting at sticking points in negotiations over a peace deal, Mr Trump said Palestinian militant group Hamas had become “very difficult to deal with” in recent weeks, suggesting this was because they only held a small remaining number of Israeli hostages.
Sir Keir has likened the plan he is working on with France and Germany to the coalition of the willing, the international effort to support Ukraine towards a lasting peace.
The Prime Minister’s official spokesman said the plan would build “on the collaboration to date that paves the way to a long-term solution on security in the region”.
He will also convene the Cabinet this week to discuss the crisis in the Middle East.
There is no indication yet of which day this week the Cabinet meeting, which gathers together the Government’s most senior ministers, will take place.
The Prime Minister is meanwhile facing calls from a growing number of MPs to recognise a Palestinian state immediately.
Palestinians have an “inalienable right to statehood”, and it is a “question of when, not if” the UK will agree to recognition, No 10 said.
More than 250 cross-party MPs have now signed a letter calling for ministers to take the step, up from 221 on Friday.
Business Secretary Jonathan Reynolds earlier dismissed the idea that there is a split at the top of Government over when to recognise a Palestinian state.
Health Secretary Wes Streeting is among those to have signalled a desire for hastened action, calling for recognition “while there’s still a state of Palestine left to recognise”, while Justice Secretary Shabana Mahmood said the Government wants to recognise a Palestinian state “in contribution to a peace process”.
Speaking to Good Morning Britain, Mr Reynolds said: “There’s no split. The whole of the Labour Party, every Labour MP, was elected on a manifesto of recognition of a Palestinian state, and we all want it to happen.
“It is a case of when, not if.”
