May is due to address Parliament later on Monday to explain her plan for Britain to adopt European Union rules on goods after Brexit, amid anger from members of Parliament in her own party who want a cleaner break and businesses who say it may still cause economic harm.
On Monday night, while addressing her backbenchers, May said without their support, Corbyn could run the country as prime minister.
He then proceeded to praise perhaps her most prominent rival, Boris Johnson, who resigned as foreign secretary last week in protest over her Brexit plan.
However, he also reportedly threatened to resign from government on a number of occasions, most recently during talks to resolve a dispute with pro-EU Conservative lawmakers who wanted a greater say for parliament over the final Brexit deal.
Born Alexander Boris de Pfeffel Johnson, he is also known for his displays of questionable sportsmanship.
The responsibility for leading the negotiations should now go to an "enthusiastic believer in your approach, and not merely a reluctant conscript", he said.
Mrs May wrote in reply that she was "a little surprised" to receive the resignation "after the comprehensive and detailed proposal which we agreed as a cabinet".
"What we are proposing is challenging for the European Union, it requires them to think again, to look beyond the positions they have taken so far and agree a new and fair balance of rights and obligations", she said.
Mr Davis's exit may embolden Brexiteer backbench MPs with concerns about Mrs May's leadership.
Jacob Rees-Mogg, the leader of a group of Brexit supporters in the Conservative Party, signalled that he would not vote for May's plan, apparently agreed by the cabinet at Chequers on Friday, if it came to a vote in parliament.
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"It does appear to me that there has been a year's worth of cloak and dagger to land us into the Chequers position", he told The Sunday Telegraph.
"And I think without David Davis there, without his imprimatur, it will be very hard for them to get the support of Conservative MPs and therefore the Prime Minister would be well advised to reconsider them".
Under Conservative Party rules, a confidence vote in a leader can be triggered if 15 per cent of Conservative lawmakers - now 48 - write a letter requesting one.
"I don't necessarily think because the British position has changed and has become much more credible, that it's the final picture, but it's important to recognise progress".
Johnson sent an incendiary resignation letter on Monday accusing May of killing "the Brexit dream" and flying "white flags" of surrender in negotiations with the European Union.
Veteran Tory Eurosceptic Sir Bill Cash told Sky that discussions were now being held as to whether Mrs May's plans amounted to a "proper Brexit".
In the Commons, Mrs May will acknowledge that there have been "robust views" around the Cabinet table and a "spirited national debate" since the 2016 referendum decision to leave the EU. "I know [new foreign secretary] Jeremy Hunt is looking forward to forging his own relationships with his U.S. counterparts". "This is the right Brexit".
"We deliver that Brexit and we do it in a way that protects jobs and livelihoods and meets our commitment to Northern Ireland".
Davis" late substitution with Dominic Raab - a hard-line Brexiteer - has clearly failed to calm a team revolt against their manager; and penalties look increasingly more likely for the United Kingdom in its Brexit negotiations.' Concludes David: "To be honest, we're now more confident of football coming home than a great deal for United Kingdom exporters and importers concluding our Brexit negotiations.