Former chief whip Andrew Mitchell is claimed to have said at a private dinner that the Prime Minister had “lost her authority” and was “weak”.
Mr Mitchell said the account of the gathering at which he was alleged to have said Mrs May could not continue was “overheated”.
Former international development secretary Mr Mitchell is alleged to have made the comments at a dinner on June 26, the day Mrs May struck a deal with the DUP to prop up her minority administration in Parliament.
A Conservative MP present at the gathering told the Mail on Sunday: “Mr Mitchell effectively said she was dead in the water. He said she was weak, had lost her authority, couldn’t go on and we needed a new leader. Some of us were very surprised and disagreed with him.”
Mr Mitchell told the Press Association: “This is an overheated report of a private dinner conversation.”
The Sutton Coldfield MP is a close ally of Brexit Secretary David Davis, who is seen as a potential successor to Mrs May, and ran his friend’s 2005 leadership bid.
Mr Mitchell did not mention Mr Davis in his comments at the One Nation Commons dining club of Tory MPs, of which he is the secretary, the Mail on Sunday said.
Meanwhile Mr Davis’ former deputy minister in the Department for Exiting the European Union used an article in the newspaper to warn that some Tories were colluding in a plot to “stymie” Brexit.
David Jones wrote that “fanatically pro-EU MPs, both Labour and misguided Tory ones, too, have hatched a cunning plot” to water down Brexit by keeping the UK in the single market and customs union.
He said: “Parliament exists to reflect and enact the will of the people - not to subvert it. But amid the heat of high summer, it looks very much as if a plot is under way at Westminster.
“A plot to stymie the clear, unequivocal wish of the British people to leave the EU, as expressed in last year’s referendum.
“And it is, I very much regret to say, a plot that may yet succeed if MPs who truly respect that referendum result allow this conspiracy to proceed.”