This could mean Brexit being delayed, a change of government or a second referendum, he said.
Appearing in the House of Commons just minutes later, a confident-sounding May defended her Brexit plan.
But the Stone MP said he was encouraged that the release of the Government's Brexit white paper had been deferred for more than a week.
Johnson had been due to host a summit about the western Balkans on Monday afternoon but was instead holed up in his official residence with close advisers, considering his position.
Johnson may be warming up his vocal chords to sing a new Brexit song that does not stick in his throat. "The trouble is that I have practised the words over the weekend and find that they stick in the throat".
Brexit negotiations with Brussels are also expected to resume next Monday.
As May attempts to unify her party behind her proposals, commentators said Trump meeting Johnson would suggest backing for his view that May's plans would inhibit Britain's ability to forge new free trade deals.
May says the plan will deliver frictionless trade with Europe and avoid a hard border between the UK's Northern Ireland and European Union member Ireland.
He wrote in The Sun that the British people "don't want some bog roll Brexit, soft, yielding and seemingly indefinitely long".
Pro-Brexit politicians commended the two ministers for their decision not to support the latest blueprint, with eurosceptic backbencher Jacob Rees-Mogg calling Mr Davis and Mr Johnons "courageous" for standing up for British voters.
If a further 47 Conservative MPs send in letters it will trigger a vote of confidence in the leader.
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Pogba also received praise from his manager Didier Deschamps , who paid tribute to the discipline he showed as France moved into their first World Cup final since 2006.
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But others argued May had adopted the most realistic plan after she was weakened in an ill-judged election previous year when she lost her parliamentary majority, meaning she has to rely on a small Northern Irish party to govern.
Lord Hague said the proposals agreed at Chequers were "the most that can be achieved within the inevitable and inescapable constraints".
Brussels has repeatedly warned it will not accept "cherry-picking" of elements of its single market. "I'm not sure. That's not on the schedule, but the president makes his own schedule".
In a letter, May told Johnson she was a "little surprised" to receive his resignation letter following productive Brexit talks on Friday. "But. who knows?", he added, wondering if Britain might reject the divorce.
Boris's resignation letter is scathing.
"It's really kind of irrelevant to what we're trying to do, and it certainly doesn't express the opinion of the British people that I've met", Ambassador Johnson said of the balloon.
Davis said his job required "an enthusiastic believer in your approach, and not merely a reluctant conscript".
'And maybe we'll speak to him when I get over there.
"It's regrettable it hasn't been resolved yet, I think there should be less hype and over spinning and let's focus on detailed negotiations".
Brexit hardliners welcomed his move, with Conservative MP Peter Bone saying: "The PM's proposals for a Brexit in name only are not acceptable".
After a dramatic day of twists and turns in Westminster, the prime minister addressed Conservative MPs for an hour, issuing a stark warning that divided parties lose elections and telling her party that "to lead is to decide".