Most seem to have concluded that the answer is not Mrs. May. There may be no immediate plot to topple her, but the prime minister’s campaign in June was so poor that few believe she will be allowed to fight the next election, which is scheduled for 2022 but could come sooner.

“It would remind voters of what she was like last time — it’s a complete nonstarter that she would carry on,” said Tim Bale, professor of politics at Queen Mary University of London, adding that her best hope was to hang on until Britain formally left the European Union in March 2019.

The next 18 months will be dominated by Brexit, which has provoked internecine warfare in the cabinet between those who want a clean break with the bloc — a so-called hard Brexit — and those who want a less radical departure, or soft Brexit, to protect the economy from a “cliff edge” change of trading rules.

Mr. Johnson, who unlike Mrs. May campaigned to quit the bloc in last year’s referendum, has positioned himself as leader of the hard-Brexit faction. That referendum was not his first big electoral victory: He was twice elected mayor of London, and is an entertaining figure in an age of celebrity politics.

Yet some see his attention-grabbing tactics as a sign of desperation, motivated by concern that his time may be running out after a decade in the public eye. With fresher faces like James Cleverly, Johnny Mercer and Tom Tugendhat emerging, Mr. Johnson’s best chance of winning the party leadership might be a quick contest before March 2019. There has been speculation that he is trying to provoke Mrs. May into firing him and making him a “Brexit martyr,” something she has so far resisted.