David Davis has topped a survey of Tory members as the candidate to replace Theresa May as leader but most want the Prime Minister to remain in post.
More than a fifth, 21%, of Conservatives want the Brexit Secretary to take over from the ailing premier, according to a study by academics.
Foreign Secretary Boris Johnson is his main rival on 17% while backbencher Jacob Rees-Mogg made a surprise entry as the third choice with 6%, the party members project funded by the Economic and Social Research Council found.
But more than a quarter, 26%, did not know or declined to say who they wanted to see as Mrs May's successor, according to the data obtained by the Observer.
And 71% said they were reluctant for the PM to quit now compared to 22% who want her to go.
Mrs May was forced to warn the Cabinet that divisions in the party could pave the way for Jeremy Corbyn to take the keys to No 10 after her top team was plunged into open warfare following a series of vicious briefings against Chancellor Philip Hammond.
Senior members of the 1922 Committee, which represents Conservative backbenchers, have indicated there is no appetite for a leadership election and that Mrs May would have their backing if she sacked plotters.
But speculation over how long she will remain in the role has continued as MPs break for the summer recess.
Anger among activists over the disastrous general election campaign has been relayed back to the party's board, according to the Sunday Telegraph.
A report was handed over that criticised "every aspect of the campaign", including the manifesto "disaster" and decision to focus on the PM, it said.
Ed Costelloe, chairman of the Grassroots Conservatives activist group, told the newspaper: "After the election can she actually recover? It would need something huge and I can't see it happening frankly.
"She is effectively crippled. She herself must know that she will not be a long-term prime minister."