The House of Commons leader widely tipped as a key contender in any future leadership race - is said to have been talking to MPs about their demands in return for support.
Toughening up her approach to trans rights is one of the key discussions.
It follows previous concerns over her comments that trans women, who are biological men, are women.
Ms Mordaunt, 51, is said to be at risk of losing her Portsmouth North seat at the next general election.
A source said that means she is open to replacing the Prime Minister if he faces a challenge after the local elections.
She is looking to talk to MPs from all wings of the Tory party, including the One Nation group of centrists.
Ms Mordaunt has a higher public profile than many of her colleagues, after the key role she played in the King's Coronation a year ago.
The insider said: "Penny has proved she is good at coronations."
Ms Mordaunt has explicitly denied being involved in any plot to oust the PM. She insisted at the weekend that she was fully behind Mr Sunak's premiership.
An ally said: "It's total hooey. Penny's getting on with the job, has been out supporting local candidates this week and is focusing on returning our candidates at the local elections."
Business and Trade Secretary Kemi Badenoch, 44, who has one of the few remaining safe seats Saffron Walden in Essex - is also tipped as a future Conservative leader.
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Denne historien er fra May 02, 2024-utgaven av Daily Express.
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