Suella Braverman has backed Foreign Secretary Liz Truss to be the next Tory leader after being eliminated from the contest to be the next prime minister.
Ex-Chancellor Rishi Sunak came first in the latest round of voting, with Penny Mordaunt second and Ms Truss third.
Ms Truss has been looking to pick up support from the right of the party in a bid to reach the final two.
Former Brexit Minister Lord Frost has urged fourth-placed Kemi Badenoch to step down to back Ms Truss.
Tom Tugendhat is the fifth candidate in the contest, which will see Tory MPs whittle the runners down to two through a series of votes, before a postal ballot of Conservative Party members will decide the winner.
Lord Frost gave the foreign secretary his endorsement before the first of three televised debates later between the candidates.
And writing in the Daily Telegraph, he suggested former equalities minister Ms Badenoch should pull out of the race “in return for a serious job in a Truss administration”.
“We need unity among free marketeers. Kemi and Suella Braverman set out convincing programmes, with differing emphases, for change,” Lord Frost said.
“But Liz’s depth of experience, her energy and ideas – as well as the simple fact she has the most votes of the three – put her in the lead.”
However, Ms Badenoch’s campaign said she “has no intention of stepping down and is in it to win”.
A spokesman for the MP said she had brought “interesting ideas and a new approach to the contest”.
The five candidates will take part in debates on Friday on Channel 4, Sunday on ITV, and Tuesday on Sky.
Former Brexit minister Steve Baker has also backed Ms Truss, and the BBC understands most of the 27 Tory MPs who voted for Ms Braverman in round two are expected to do the same.
Attorney General Ms Braverman had styled herself as the only “authentic” Brexiteer candidate in contention, pointing to her resignation as a Brexit minister under Theresa May over the former prime minister’s deal.
She told the BBC that Ms Truss was the best candidate to deliver upon “our Brexit promises”, and would take a “firm line” on illegal migration, freedom of speech and cutting taxes.
Mr Sunak is still in pole position after winning 101 votes in Thursday’s ballot, followed by trade minister Ms Mordaunt, who gained the most momentum with 83 votes.
Ms Truss secured 64 votes.
Like Ms Badenoch, who came fourth with 49 votes, foreign affairs committee chairman Tom Tugendhat – fifth on 32 – said he is determined to stay in the contest.
The next round of voting, when the candidate with the least votes is eliminated, takes place on Monday.
The final result will be announced on 5 September, when Boris Johnson will leave office.
Mr Sunak has now topped the first two ballots of MPs, with other candidates scrambling to join him in the final run-off.
His decision to quit as chancellor last week was one of the first in a wave of ministerial resignations that forced Mr Johnson to stand down as Tory leader.
Allies of Mr Johnson have accused Mr Sunak of orchestrating the prime minister’s downfall and have been fiercely critical of tax hikes he introduced as chancellor.
A member of Mr Sunak’s team said the former chancellor would “keep going with solid messaging on the economic agenda”, claiming other candidates had backed away from their pledges.
Following Thursday’s vote, Welsh Secretary Robert Buckland became the latest senior Tory MP to back Mr Sunak, telling Talk TV he was the best person to deal with the economic challenges facing the country.
Ms Truss’s team said Thursday’s result showed the foreign secretary was “attracting a wide range of supporters from across the Conservative Party”.
Following the launch of her campaign on Thursday, a senior member of Ms Truss’s campaign urged backers of Ms Badenoch and Ms Braverman to “recognise the reality of the situation” and unite behind the foreign secretary.
But a campaign source for Ms Badenoch said she was “happy” with the result, given “this time last week she hadn’t decided if she was running”.
“Now is the time for change and I am looking forward to continuing to make that case in the TV debates,” Ms Badenoch wrote in a tweet.
She said she was “disappointed” that Ms Braverman had backed Ms Truss, telling LBC “the tough thing to do is to take a risk and try something different”.
There had been speculation that Mr Tugendhat might drop out of the contest. But in a tweet, he said he would stay and take part in the debates to put his “vision for Britain forward to the public”.
The backbencher, who won five fewer votes in the second round than in the first, said his campaign for “a clean start” continued, adding: “We need trust back in our politics.”
Source: BBC