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.