The donor's intervention comes as Grant Shapps, the former Conservative Party chairman who resigned following the Tatler Tory scandal, claimed that 30 MPs and "one or two" cabinet ministers had agreed to sign a letter calling on Mrs.
Conservative Party members then vote for their preferred candidate on a "one member, one vote" basis, with the victor becoming the new leader.
Veteran backbencher Michael Fabricant said of Shapps: "I wouldn't buy a used auto from one embittered colleague - let alone take advice from him about who should be PM".
First there was the prankster, theatrically handing her a notice that she was sacked; then a persistent cough turned her delivery into a croak that was painful to the ear.
In an attempt to reboot her leadership, she apologized for the election fiasco in a speech to the Tory party conference on Wednesday.
The comedian had signed the document with the name of Boris Johnson - May's foreign secretary, once thought to have had leadership ambitions.
To trigger a formal leadership challenge, 48 Conservative lawmakers need to write to the chairman of the party's so-called 1922 Committee.
During her speech, May struggled with a cough and a hoarse voice that forced her to pause repeatedly.
Damian Green said it was "nonsense" to suggest she should go over the speech.
David Lidington, the Justice Secretary, told i: "Most people out there in the real world know they have to get up and go to work coping with a bad throat or a cold from time to time". She said the government wants Brexit negotiations to succeed, but is also preparing for a scenario in which they could fail.
He said the "overwhelming majority" of Tory MPs - including the "entirety" of the Cabinet - wanted her to carry on.
May's second betrayal came from the audience, which concealed a man waiting to rush the stage to give May a fake government pink slip dismissing her from her job.
He was angrily denounced by MPs loyal to the prime minister, with Mr Walker dismissing his supporters as a "coalition of the disappointed" who had been overlooked for promotion.