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 they 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 did not 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.
Mr Stewart also received a boost when Cabinet Office Minister and de facto deputy prime minister David Lidington backed him for the top job.
Mr Lidington, who had supported Mr Hancock before he quit the race, said Mr Stewart was the right person to best connect the Tory party with the country.
The Cabinet Office Minister told a pro-Stewart gathering in London: “I think there is a yearning in this country for political leaders who tell it straight to people.
“What Rory has done in his campaign is to demonstrate that there are no no-go areas in this country for him or for the party which he aspires to lead.”
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.”
Meanwhile, current number two Jeremy Hunt stood out as the only candidate to endorse US President Donald Trump’s retweet of comments attacking London Mayor Sadiq Khan.
The Foreign Secretary said he agreed “150%” with the “sentiment” of the Katie Hopkins’ comment about “Londonistan” retweeted by the president – despite it being labelled racist.
After the hustings, Mr Stewart tweeted: “I 100% disagree with both the language and the sentiment of the last sentence of this tweet.
“Can all candidates please confirm the same.”
Mr Javid called the comments “unbecoming”, Michael Gove said they were “a mistake” and Dominic Raab called them “not helpful”.
I 100% disagree with both the language and the sentiment of the last sentence of this tweet. Can all candidates please confirm the same https://t.co/cx6gapQExQ
— Rory Stewart (@RoryStewartUK) June 17, 2019
In a dig at front-runner Mr Johnson’s non-appearance, former Brexit secretary Mr Raab used his hustings slot to describe the event as an “essential gauntlet” to be run, and said he was “quietly confident” he would get through to the next round.
Mr Raab said a “united Cabinet” would allow him to achieve Brexit, claiming: “these negotiations broke down because ultimately the Prime Minister and the Cabinet were not willing to hold the line”.
Environment Secretary Mr Gove focused on the Labour leader, claiming he is the only Tory leadership candidate able to “strike fear” into the heart of Jeremy Corbyn.
He said: “I don’t believe that any of the other candidates will strike fear into his heart in quite the way that I have proven that I can.”
About 160,000 Tory members across the country will decide the next prime minister, from the final two candidates chosen by Conservative MPs.
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.
With Matt out, I'm still looking for the same things:
Energy, intelligence + enthusiasm with a clear vision for delivering Brexit + for the country beyond it.
Someone who really gets the Union, with ideas to strengthen + secure it.
Tomorrow I'll be voting for @RoryStewartUK
— Paul Masterton MP (@PM4EastRen) June 17, 2019
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 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.”
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?”