More candidates are expected to enter the Tory leadership race to challenge frontrunner Boris Johnson for the Conservative Party crown.
Work and Pensions Secretary Amber Rudd ruled herself out of the contest, but speculation mounted that Health Secretary Matt Hancock would announce on Saturday that he was in the running.
More than a dozen Tories are understood to be considering a bid, with Foreign Secretary Jeremy Hunt indicating he will be in the race and former work and pensions secretary Esther McVey putting herself forward “as a future leader”.
The new Tory leader looks set to take over as prime minister at the end of July after Theresa May finally laid out a timetable for her exit from Downing Street.
Former foreign secretary Mr Johnson, who has emerged as the bookies’ favourite, stressed he would be prepared to back a no-deal departure to ensure the UK leaves the EU on October 31.
The timetable for the contest will see nominations close in the week of June 10, with MPs involved in a series of votes to whittle down what is set to be a crowded field to a final two contenders.
Tory party members will then decide who wins the run-off.
Ms Rudd told the Daily Telegraph: “I am conscious the Conservative Party wants someone who they believe is very enthusiastic about Brexit.
“There are all sorts of plans I would like to have when we do leave the European Union but I don’t think it is my time at the moment.”
The Cabinet minister made it clear she would not have a problem working with Mr Johnson in Government again.
However, Prisons Minister Robert Buckland expressed concern about a no-deal exit from the EU.
He told BBC2’s Newsnight: “My message to candidates is to think very carefully before hurtling away to a position of ‘no deal’ and not just the self-interest of a leadership competition.
“For them to box themselves into their red lines is not the wisest way to go into this leadership election.”
Tory MP George Hollingbery said Mrs May was disappointed in herself.
He told the BBC: “Her sense of duty drives her from her very core.
“For her she is so disappointed in herself I believe, because she has not delivered what she ought to have delivered for the benefit of the country and she feels she should have done better. But leaving a legacy is not her style.”
Digital Minister Margot James has raised questions over how representative the Tory membership set to elect the new party leader is of Conservative voters.
Ms James told Channel 4 News: “Effectively, the prime minister is going to be chosen by roughly 100,000 people, assuming we go that far into the contest, who don’t… some would argue they don’t really represent the majority of Conservative voters, let alone the country as a whole.”
Labour leader Jeremy Corbyn said Mrs May’s replacement should call an immediate general election.
US president Donald Trump, who makes a state visit to the UK in early June, praised Mrs May.
He said: “I feel badly for Theresa. I like her very much. She is a good woman. She worked very hard.
“She is very strong. She decided to do something that some people were surprised at. Some people weren’t.
“It’s for the good of her country. But, I like her very much.
“In fact, I’ll be seeing her in two weeks.”
Mr Hunt, MP for South West Surrey, heavily hinted he would join the race to replace Mrs May, telling his local newspaper the Farnham Herald: “I’ll make the announcement on my own candidacy at the appropriate time.”
Hosting a radio call-in on LBC, Ms McVey said: “I’ll put my hands up here, I better declare an interest straight away. I have put myself forward as a future leader.”
And Sir Graham Brady quit as the leader of the 1922 Committee – a position which gave him a significant role in the Prime Minister’s departure – in order to consider a leadership bid.
Environment Secretary Michael Gove – who stood in the 2016 leadership race and may consider another bid – said Mrs May “deserves our respect and gratitude”.
Home Secretary Sajid Javid, who had a “frank” discussion with Mrs May about her deal on Thursday, said “nobody could have worked harder or had a greater sense of public duty”.