If she falls, any Conservative MP elected by the party as leader would automatically become prime minister.
Here are the main names mooted by the British media:
Sajid Javid
A former investment banker and the son of a Pakistani bus driver, the 49-year-old Sajid Javid is the face of a modern, multi-cultural and meritocratic Britain.
On the economically liberal wing of the Conservative Party, Javid voted for Britain to stay in the European Union in 2016 but has since backed the Brexit cause.
Since being appointed interior minister in April, he has earned respect for his handling of a scandal over the treatment of the children of Caribbean immigrants known as the Windrush generation.
Boris Johnson
A former mayor of London, "Boris" or "BoJo" was a key figure behind the 2016 Brexit campaign but failed in his bid to become prime minister in the aftermath as an ally pulled his support at the last minute.
May appointed Johnson foreign minister but he quickly drew attention for all the wrong reasons, including a series of diplomatic gaffes. He became a trenchant critic of the government's Brexit strategy before resigning in July.
Charismatic and popular in some circles, the 54-year-old has also earned plenty of enemies among Conservatives for his behaviour.
Asked on the BBC's Andrew Marr Show on Sunday whether he was planning a leadership bid, he avoided the question but dismissed as "nonsense" reports that he was already offering ministerial posts to supporters.
Amber Rudd
Elected to parliament in 2010 after a career in business and financial journalism, she has been a staunch supporter of May in her rise to power.
Rudd was given May's old job of interior minister when the latter entered Downing Street in 2016, but was forced to resign in April to protect the prime minister's reputation over the Windrush scandal.
She has been recently welcomed back into the cabinet as work and pensions secretary.
The hard-working, 55-year-old could, however, be undermined in any leadership bid by her support for Britain remaining in the EU.
Michael Gove
Brexit campaigner Gove initially supported Johnson's leadership bid in 2016 but at the last minute announced his own intention to run, causing both men to lose out to May.
After a year in the political wilderness, he was appointed environment minister in June 2017 and has stayed in the headlines with a series of eco-friendly policy announcements.
As the leading eurosceptic in government, the cerebral 51-year-old's support for May's Brexit deal had been viewed as crucial.
Dominic Raab
The 44-year-old was appointed Brexit minister in July, to replace David Davis who quit over May's deal, only to resign himself in November over the latest agreement struck with Brussels.
Raab has a black belt in karate and is a former international law specialist, and has not ruled himself out of running for May's job.
Jeremy Hunt
The current foreign minister supported remaining in the EU but has been highly critical of what he calls the "arrogant" approach taken by Brussels in the negotiations.
A former businessman who also speaks fluent Japanese, the 52-year-old is a resilient politician, having headed up the National Health Service for six years during a funding crisis.
Penny Mordaunt
The 45-year-old is a rising Conservative star and bona fide Brexit supporter. The international development minister has reportedly been critical of May's Brexit deal in private although she has remained outwardly loyal.
Esther McVey
The 51-year-old is a former work and pensions minister who quit the government in November in protest against the Brexit deal. She has said she would run if asked.
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