Former Foreign Secretary Mr Johnson claimed his Brexit dream "is dying" in his scathing resignation letter.
The prime minister hammered out a compromise with her deeply divided Cabinet in an all-day meeting at Chequers on Friday, but after consulting friends and allies since, Johnson decided he could not promote the deal, according to the report.
May's proposals for a future European Union relationship after Britain departs from the bloc next March had taken two-years of internal government wrangling to agree, but within 48 hours Johnson and Davis had resigned saying they could not back the plans.
He added: "Since I can not in all conscience champion these proposals, I have sadly concluded that I must go".
Labour leader Jeremy Corbyn said Mrs May should make way for a Labour administration if her Government can not get its act together quickly.
"The prime minister knows it's a proposal and not an end agreement".
May imposed on her cabinet on Friday.
British and European Union officials are hoping to strike a deal on the terms of Britain´s withdrawal and agree to a plan for future trade ties in time for an European Union summit in October.
Speaking at the Council on Foreign Relations in New York, Barnier said he looked forward to a "constructive discussion" with the United Kingdom after the white paper on Brexit is published on Thursday.
She also said that the government had no intention of extending Article 50, which is a notice of intention to leave the European Union, and that there will not be a second national vote on this Brexit deal.
"It seems to me we're giving too much away, too easily, and that's a unsafe strategy at this time", Davis said in a BBC radio interview Monday morning.
However, at a meeting with her Conservative Party lawmakers, she was cheered and applauded by many as she warned them that internal squabbling could pave the way for socialist opposition leader Jeremy Corbyn to take power instead.
But leading pro-Brexit legislator Jacob Rees-Mogg said "I don't think a no-confidence vote is immediately in the offing".
In her statement to Parliament, May attempted to give an impression of business as usual, saying she would present a detailed paper onThursday on the government's negotiating position for a post-Brexit relationship with the EU.