LONDON—Prime Minister Theresa May is facing fresh questions about whether she can stay in office long enough to see Brexit through, as lawmakers enter another critical week of decisions over the U.K.’s departure from the European Union.

Mrs. May is encountering growing dissatisfaction from her Conservative Party colleagues over her handling of Brexit after a bruising encounter last week with EU leaders, who agreed to delay Brexit.

Pro-Brexit lawmakers have described Mrs. May’s request for a delay as a national humiliation. Speculation packed the nation’s newspapers on Sunday that her cabinet was on the brink of forcing her from office, though they couldn’t agree on who her replacement would be.

Ejecting Mrs. May would be tricky, and she has fought off past attempts to unseat her. If she did leave office, her successor would face the same challenge of leading a divided party without the parliamentary majority needed to force their vision of Brexit through, meaning the uncertainty clouding the U.K.’s future would persist.

“Changing the players doesn’t solve the problem,” Treasury chief Philip Hammond said Sunday in an interview with Sky News, in which he denied being part of a plot to topple Mrs. May and install David Lidington, her de facto deputy, as leader instead.