The interview was published as May greeted Trump at a lavish welcome reception at Blenheim Palace, the birthplace of Winston Churchill. Johnson was slightly more diplomatic in his resignation letter, warning May that Britain was "headed for the status of a colony" within the EU.
The Chequers policy, which secured Cabinet agreement last week before sparking senior resignations, includes being tied to the European single market for goods under a "common rule book" and close customs arrangements as part of a new UK-EU free trade area.
Hammered out at Theresa May's Chequers retreat, the plan has upset Tory Leavers, who fear it doesn't amount to a "hard Brexit", with David Davis and Boris Johnson quitting their ministerial posts in protest.
Trump attended a black-tie dinner at Blenheim Palace with May on Thursday evening, where the TV network ITV reports May planned to pitch the president on a free-trade agreement between the United Kingdom and the U.S. after Brexit. "The direct consequences of that will be Prime Minister Corbyn".
Asked if May should remain in power, Trump said "that's up to the people".
"I think it changed the fabric of Europe and, unless you act very quickly, it's never going to be what it was and I don't mean that in a positive way", he cautioned.
She praised the friendship between the two allies, glossing over Trump's previous remarks that Britain was a "hot spot" in turmoil over Brexit.
A 98-page report setting out the Government's demands to Brussels would mean European Union workers will still be allowed rights to travel to Britain temporarily.
Mrs May's Conservative Party only has a majority in Parliament with the support in key votes of the 10 MPs from Northern Ireland's Democratic Unionist Party, so any split raises questions about whether her plan could survive a Commons vote - and has also led to renewed questions about whether she will face a challenge to her position.
Caulfield said in her resignation letter that "I cannot support the direction of travel in the Brexit negotiations, which in my view do not fully embrace the opportunities that Brexit can provide". "That is exactly what we will do".
Khan gave protesters permission to fly the 20-foot-tall balloon depicting Trump as an angry baby in a diaper.
Europe s chief negotiator Michel Barnier said Tuesday talks will resume next Monday as scheduled with the new Brexit minister, while refusing to comment on the political infighting consuming London. The U.K. would agree to abide by the EU's rulebook for goods and agricultural products in return for not losing its open customs borders with the union.
Brexiteer Andrea Jenkyns, a vocal critic of Mrs May, said it was time to "put country first".
Free trade would not apply to services, which account for 80 per cent of the British economy.
May is promoting the "soft" Brexit plan, which would keep strong economic ties with the European Union.
"We are coming out of the European Union and we are going to continue to grow our economy, do deals around the world and importantly for a constituency like mine end free movement".
And he said it was "impossible" to say whether Brussels was likely to accept the proposals.
Pro-Brexit Conservative lawmaker Jacob Rees-Mogg colourfully described the plan as "the greatest vassalage since King John paid homage to Phillip II at Le Goulet in 1200".