May's leadership shattered after the United Kingdom snap parliamentary election in June, which, despite its initial goal to unify the political leadership ahead of Brexit negotiations, resulted in a hung parliament and prompted the Conservatives to agree on a deal with the Democratic Unionist Party (DUP). Under the party's rules, a leadership race can be triggered if at least 48 of those MPs express their support.
But the former party chairman, Grant Shapps, said Ms May's leadership should now be challenged.
In a keynote conference speech where the protester and a coughing fit brought her words nearly to a halt, Mrs May won over many members in the hall by promising to reinvigorate the party and offering pledges to younger people and families alike.
In one of the most freakish British political speeches in memory, Ms May's address on Wednesday to her annual conference was ruined by coughing fits, a comedian handing her a bogus employment termination notice, and by letters falling off the slogans on the set behind her.
He said the "overwhelming majority" of Tory MPs - including the "entirety" of the Cabinet - wanted her to carry on.
Mr Shapps, who has claimed to have the backing of around 30 MPs - with some Cabinet members also privately offering support - said the demands for an election were growing.
It is not known whether Boris Johnson, who has been the subject of fevered leadership speculation amid his perceived disloyalty on Brexit in recent weeks, was one of the ministers who called Mrs May.
But the Prime Minister, who did a round of several dozen radio and TV interviews yesterday before attending party receptions, had a bad cough and struggled through much of the rest of her hour-long speech, her voice weakening noticeably towards the end.
Writing on his Twitter feed shortly afterwards, Brodkin said: "Hi @BorisJohnson, I gave Theresa her P45 just like you asked".
They are also terrified that another leadership contest could lead to a resurgent Labour Party under Jeremy Corbyn winning power.
Shapps has been vocal in his concerns about the prime minister's position recently and this morning confirmed he was behind attempts to persuade her to step down.
"It's been really hard for her but we really are behind her".
Environment Secretary Michael Gove, a prominent Brexit campaigner, said Friday that May had "shown tremendous grace and grit in the course of this week". But May's distress appeared to rouse the delegates, who gave her repeated standing ovations - and when she could not speak, urged her on by shouting "Come on, prime minister!".