Tory leadership hopefuls Rory Stewart and Sajid Javid believe they have the required number of supporters to survive Tuesday’s second round of voting.
Mr Stewart managed to secure just 19 votes in the first ballot and Mr Javid had 23 – both short of the 33 required to stay in the race after the second vote.
But the pair told journalists at a special hustings in Westminster that they were confident of remaining in the contest to be the next prime minister.
They are a long way short of frontrunner Boris Johnson, who picked up 114 votes last week and has since been boosted by the support of former leadership contender Matt Hancock.
Former foreign secretary Mr Johnson was not expected to appear at the hustings event for political journalists and was notably absent from the first TV debate on Channel 4 on Sunday.
But his campaign continued to gain ground with the support of Mr Hancock, which came as a blow to Environment Secretary Michael Gove – who had courted his endorsement.
At the hustings, Mr Stewart said he had the necessary 33 backers to make it through the second round of voting in the contest “if they do what they say”.
The International Development Secretary suggested he was the one to beat Mr Johnson – and accused the former foreign secretary of making different promises to different MPs.
“Who is going to be nimble enough, who has the style, who has the approach, who has the way of dealing with the public… I don’t think the answer is going to be pre-scripted answers,” he said.
Home Secretary Mr Javid said he was “extremely confident” of getting the required 33 votes.
“I think there is a growing feeling in the party that when we get to the final two we should have a robust debate between two credible change candidates,” he said.
“If we don’t get change, people will vote for change in the form of Jeremy Corbyn.”
Former Brexit secretary Dominic Raab used his hustings slot to describe the event as an “essential gauntlet” to be run, in a dig at Mr Johnson’s non-appearance.
Around 160,000 Tory members across the country will decide the next prime minister from the final two candidates chosen by Conservative MPs.
With Mr Johnson more than 70 votes ahead of his nearest rival Jeremy Hunt in the first round of voting, the contest now appears to be about who will face him in the final two.
Explaining his decision to back Mr Johnson, Mr Hancock said: “Having considered all the options, I’m backing Boris Johnson as the best candidate to unite the Conservative Party, so we can deliver Brexit and then unite the country behind an open, ambitious, forward-looking agenda, delivered with the energy that gets stuff done.”
Writing in The Times, he said he believed Mr Johnson had a “unique personality” which would bring the Tories together behind a Brexit deal, adding: “We need that unity in the Conservative Party, and then in the country. Let’s move forward.”
Mr Gove admitted he was disappointed by Mr Hancock’s decision.
“He is a friend of mine and I know that over the course of the weekend he had a very tough decision to make,” he told BBC Radio 4’s Today programme.
“Without going into private conversations, I know that he was alternating between supporting Boris and supporting me.
“He felt that we were the two strongest candidates in the race.”
The Environment Secretary, who finished third in the first round of voting, appeared to concede that Mr Johnson was assured of a place in the final two, but argued that his opponent should be ready to lead the country.
“We need to make sure that he is tested and that we have two candidates who go forward – if Boris is one of them – who we know are capable of being prime minister from day one,” he said.
Mr Gove said his experience running three government departments showed he was “ready to take control of the ship of state and steer it safely through the difficult waters ahead”.
The 20 MPs who backed Mr Hancock in the first round of voting will now have to choose their new candidate, and not all will follow the Health Secretary to Mr Johnson’s camp.
Former Hancock supporter and East Renfrewshire MP Paul Masterton said he would now back Rory Stewart.
While not facing journalists’ questions, Mr Johnson chose to use his column in The Daily Telegraph to announce plans to extend full-fibre broadband to every home in the country by within five years, nine years ahead of the Government’s 2033 target.
“A fast internet connection is not some metropolitan luxury. It is an indispensable tool of modern life,” he said.
“It is therefore a disgrace that this country should suffer from a deep digital divide, so that many rural areas and towns are simply left behind.”
Justice Secretary David Gauke, who backs Mr Stewart, mocked Mr Johnson’s repeated spending commitments.
He tweeted: “I’m the last person to want to curtail the leadership race. But every Telegraph column by Boris Johnson increases borrowing by £ billions.”
If Boris wins, good luck to whoever becomes his Chancellor. It would be a noble act of self-sacrifice to accept the job. Who’d do it? 🤷♂️
— David Gauke (@DavidGauke) June 17, 2019
In a veiled reference to reports suggesting that Mr Hancock could have his sights on 11 Downing Street under a Johnson premiership, Mr Gauke added: “If Boris wins, good luck to whoever becomes his Chancellor.
“It would be a noble act of self-sacrifice to accept the job. Who’d do it?”