Penny Mordaunt has become the first Conservative MP to say they will run to replace Liz Truss as prime minister.
The Commons leader said she was standing to “unite our country” and win the next general election.
Rishi Sunak is yet to confirm he is standing, but he has already amassed 87 of the 100 endorsements from MPs he would need to take part.
Boris Johnson has not ruled out a comeback, months after being forced out of No 10 after a Tory revolt.
The former prime minister – who is currently out of the country – has 44 MPs backing him. Ms Mordaunt currently trails both Mr Johnson and Mr Sunak, with only 21 public supporters.
International Trade Minister James Duddridge, a former parliamentary aide to Mr Johnson, has told the BBC: “Boris is coming, and he has the momentum and support. He is coming home and is up for it.
“He is the only election winner we have that has a proven track record in London, on Brexit, and in gaining the mandate we have now.”
Ms Mordaunt is currently Commons leader and was briefly defence secretary under Theresa May.
The 49-year-old Portsmouth North MP backed Brexit and has worked hard to build up her profile with party activists, but did not get enough support from MPs to reach the members’ ballot last time.
Tory party bosses have decided contenders will need 100 nominations from fellow Conservative MPs, a much higher threshold than normal, meaning no more than three MPs will be able to stand.
Defence Secretary Ben Wallace has ruled himself out of the contest, adding he is “leaning towards” backing Boris Johnson.
He told reporters Mr Johnson had a “mandate” from the 2019 general election and “could win the next election”.
Mr Sunak has won a high-level endorsement of his own, with former chancellor and home secretary Sajid Javid backing his leadership bid.
Tory MP Angela Richardson said on Friday she had signed Mr Sunak’s “nomination papers for Conservative Party leader”.
Ms Richardson, who has already declared her support for Mr Sunak, described this as a “positive end to an extraordinary week in Westminster”.
Earlier this week, the leader of the Commons Penny Mordaunt told MPs the prime minister was not “under her desk” hiding when she stood in for Ms Truss at the despatch box.
It sounded more mischievous than helpful and Ms Mordaunt’s own ambition to sit behind the No 10 desk has never been in doubt.
She is hoping for early momentum by breaking cover first. Rishi Sunak currently has the most publicly declared support among Tory MPs and the former chancellor is expected to announce he’s running soon.
They will both pitch themselves as unifying candidates.
With the threshold for nominations set so high, a maximum of three candidates can reach the first ballot on Monday.
The big question heading into the weekend is whether Boris Johnson attempts to boomerang back into Downing Street.
The former prime minister has already got a couple of Cabinet endorsements but other Tory MPs are horrified at the idea of Mr Johnson making a comeback.
It’s a prospect that splits Conservative MPs between delight and despair. The chance of a backroom deal and simple coronation seems to be receding as the rival campaigns fire up.
If Mr Johnson did make a comeback it would be unprecedented in modern British political history, coming three months after he was forced out by his own MPs following a string of scandals.
Tory MP Jesse Norman, one of scores of ministers to quit his government over the summer, said bringing him back would be an “absolutely catastrophic decision”.
And former Conservative leader Lord Hague told Times Radio it “would be going round in circles” and “could become a death spiral” for the party.
“I think it’s possibly the worst idea I’ve heard of in the 46 years I’ve been a member of the Conservative Party,” he said.
Polling suggests he remains popular with Tory members, but another tilt at the leadership could prove divisive among Tory MPs, some of whom have said they would quit politics if he came back.
‘Indicative’ MP ballot
Chancellor Jeremy Hunt has ruled himself out of the race, along with Security Minister Tom Tugendhat and former cabinet minister Michael Gove.
Former Home Secretary Suella Braverman and International Trade Secretary Kemi Badenoch are thought to be considering bids.
Contenders have until 14:00 BST on Monday to find 100 backers. If three reach the threshold, Conservative MPs will knock out one contender in a ballot on the same day.
MPs will hold an “indicative” ballot of the final two, with the winner then decided in an online vote of party members, to finish next Friday.
