To proceed further, candidates will then need 33 MPs - 10 per cent of Tory MPs - to vote for them in the second round on June 18.
In the 2016 referendum, 52 percent of voters chose to leave the European Union, while 48 percent wanted to remain.
Britain's opposition Labour Party narrowly held on to a parliamentary seat in eastern England on Friday, seeing off a challenge from Nigel Farage's insurgent Brexit Party to win by less than 700 votes.
May will remain as prime minister for a few weeks while the party picks a successor, who will become Britain's next prime minister.
While nominations will take place next Monday, 11 Members of Parliament have already indicated their intention to replace Mrs May, with front-running backbencher and former foreign secretary Boris Johnson officially throwing his hat into the ring five days ago.
The move will pave the way for a leadership battle for her successor who will try to deliver Brexit after May failed to get her deal through Parliament.
The new Brexit Party, which came first in May's European Parliament election, beat out the Conservatives by more than seven per cent (28.9 to 21.4 per cent) and the Liberal Democrats, who came fourth with 12.3 per cent, both long-established parties.
British Prime Minister Theresa May's time as Conservative Party leader ended today, not with a bang but a whimper.
Hard-core Brexiteers in the British leadership contest say they would rather take Britain out of the bloc with no deal than countenance a further delay.
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This could happen either because the Democratic Unionist Party calls a halt to its support in the House of Commons for the minority Conservative government, or a handful of Conservative MPs decide to oppose May's successor, said Begg, adding "Neither is implausible".
May formally relinquished her leadership in the private letter to her party on Friday.
She is stepping down amid a growing dispute with Chancellor Philip Hammond over her plans to leave with a series of big spending announcements, including a multibillion-pound overhaul of England's schools and colleges, according to the Financial Times.
"Obviously there is some very important business to then continue for the new prime minister, not least the Brexit process", Antony Phillipson, the UK's North American trade commissioner and consul general, told Cheddar in an interview Friday.
None of them have actually said that the United Kingdom is going to ditch the withdrawal bill, start again, try and do a better deal, because that's of course what the Tory Party wants, and they are actually walking away from this aspect of a no deal Brexit, which would be the most beneficial option for the United Kingdom to actually make Brexit mean Brexit.
Johnson is one of several contenders - including Environment Secretary Michael Gove, Foreign Secretary Jeremy Hunt and Health Secretary Matt Hancock - promising to go back to Brussels and make changes to the Brexit deal.
"I am now ahead of Boris and the other candidates", Mr Stewart said.
Whether it is Johnson, Hunt or one of the other contenders, political experts view Brexit as a poisoned chalice.
May will remain prime minister until a new leader is chosen, likely in late July, but has relinquished control over the direction of Britain's tortuous departure from the European Union.