The second round of voting left Rishi Sunak in the lead with 101 votes, Penny Mordaunt in second on 83 – gaining more votes than any other candidate – and Ms Truss in third on 64. Mrs Badenoch received 49 votes, and Tom Tugendhat 32.
There have been concerns among the Right of the Conservative Party that failing to fall in behind one candidate could prove a strategic error and risk leaving two more centrist MPs in the final run-off.
Writing for The Telegraph, Lord Frost said: “We need unity among free marketeers. Kemi and Suella Braverman set out convincing programmes, with differing emphases, for change.
“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.
“Free marketeers’ strength is that they have principles and convictions. Rightly so. But that can also create unnecessary dissension. It is now time for pragmatism. I urge Kemi to stand down in return for a serious job in a Truss administration.”
Mrs Badenoch’s spokesman on Thursday night said she had “no intention of stepping down”.
Lord Frost raised concerns about the prospect of Ms Mordaunt becoming prime minister, accusing her of being “absent on parade” when they worked together on delivering Brexit and saying no Government of hers could “succeed”.
Ms Braverman, the Attorney General, had the backing of prominent Brexiteers but was forced out of the contest after getting just 27 votes.
On Thursday night, she argued that Ms Truss was now the best-placed candidate to defeat Labour, telling The Telegraph: “Liz will win us the next election.
“She’s the best person to deliver on the opportunities of Brexit, will deliver a Conservative agenda on the economy, defend free speech, champion equality of opportunity and take a robust line on illegal immigration.”
In a further boost for Ms Truss, the 60-strong European Research Group of Conservative MPs was on Thursday night told by Mark Francois. its chairman, to vote for the Foreign Secretary.
In a letter seen by The Telegraph, Mr Francois – who has already declared for Ms Truss – said that, at a meeting of the ERG on Wednesday, “we did overwhelmingly agree that there were two candidates who we felt stood out among all the others, namely Suella Braverman and Liz Truss”.
He added: “While nearly all members of the group were backing either one or the other, we also agreed collectively that, if only one got through to the next round, we would then row in behind that candidate, whichever one it might be.”
Mr Francois said that, “as a result” of Ms Braverman backing Ms Truss, “I do hope that all of our group will now feel able to unite firmly behind Liz, to help deliver the vital Northern Ireland Protocol Bill, expand on the opportunities offered by Brexit and secure the future of the Conservative Party”.
Steve Baker, the former chairman of the European Research Group of Tory MPs, also backed Ms Truss, telling The Telegraph: “Suella has my complete loyalty. What she has decided, I will support.”
Earlier in the day, after getting knocked out of the contest, Ms Braverman indicated that she would back the candidate who came closest to agreeing to leave the European Court of Human Rights (ECHR) as well as building on Brexit and cutting taxes.
A source close to Ms Truss said her position was not to support withdrawal from the ECHR, though she would consider the move “in extremis” if reform was not delivered.
Mrs Badenoch said she was “disappointed” that Ms Braverman had not backed her, telling LBC: “Suella and I are friends, I’m friends with Tom Tugendhat as well, Rishi and Liz were my senior ministers, so it’s all very close to home.
“We’re all friends, so every time someone supports one person there’s another person that’s been let down. But I’m in it to win it.”
Mrs Badenoch, who has the backing of Michael Gove, the former communities secretary, gave no indication that she was considering withdrawing from the race.
On Friday, the first televised debate will take place, hosted by Channel Four, with all five remaining candidates expected to take part. Another debate will take place on ITV on Sunday night.
The next round of voting will be held on Monday, with the worst-performing candidate knocked out. The same will happen on Tuesday and again on Wednesday, after which the two remaining candidates will be put to the Tory membership.
Who they select as the next Tory leader – and therefore the next Prime Minister – will be announced on Sep 5, with the winner entering Number 10 on Sep 6.
Mr Sunak is within touching distance of reaching the final two. He secured 101 votes, with 120 guaranteeing a spot in the membership ballot.
Ms Mordaunt picked up more MP votes between the first and second round, with 16, and her team hope she has the momentum to reach the final two.