The news comes in the wake of Mrs May’s failure to persuade the EU to accept her Brexit vision.
Chief negotiator Michel Barnier ruled out her plan – for Britain to collect EU tariffs on goods passing through the country – on Thursday.
An anonymous source told the Times an autumn Tory leadership election was “nailed on”, adding: “The Europeans have her written off…she’ll be out by October”.
Andrew Bridgen, the Brexiteer Tory MP for North West Leicestershire, held a meeting of his local party on Thursday night at which a motion of no confidence in the Prime Minister was endorsed.
Meanwhile the Brexit Delivery Group, counting both Tory Remainers and Leavers among its number, hopes Britain can exit the EU without the party imploding.
Mrs May’s efforts to bypass Brussels and deal directly with EU nations fell on deaf ears this week.
The PM was warned to avoid a hard Brexit by the Austrian chancellor, Sebastian Kurz, and his Czech counterpart Andrej Babis, before departing for her summer holiday.
It comes as a poll revealed 72 per cent of voters do not have confidence in her ability to strike a good deal with Brussels.