Tory leadership rivals are scrambling to catch former chancellor Rishi Sunak after he won the second round of voting to choose Boris Johnson's successor.
Five candidates remain after Attorney General Suella Braverman was knocked out in Thursday's vote.
Mr Sunak won 101 votes, Penny Mordaunt gained ground on him, with 83 votes, and Liz Truss came third with 64.
Ms Truss's chances were boosted on Thursday evening when Ms Braverman backed the foreign secretary.
Former Brexit minister Steve Baker has also endorsed Ms Truss, and the BBC understands most of the 27 Tory MPs who voted for Ms Braverman are expected to do the same.
Former equalities minister Kemi Badenoch - who came fourth with 49 votes in Thursday's vote - and foreign affairs committee chairman Tom Tugendhat - who was fifth on 32 - both say they are 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 five remaining candidates will take part in televised debates on Friday on Channel 4, Sunday on ITV, and Tuesday on Sky.
Tory MPs are aiming to whittle down the field to two by the end of next week, when it will be thrown over to a postal ballot of Tory members to choose the winner, who will become prime minister.
The 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 the contest increasingly looking like a scrap to see who will 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.
The leadership contest became increasingly fractious on Thursday, as allies of the candidates launched personal attacks on their rivals.
Ms Mordaunt batted away criticism from former Brexit negotiator David Frost, who questioned her competence to be prime minister.
He said Ms Mordaunt - a supporter of Brexit - "wouldn't always deliver tough messages to the European Union" when he worked with her during negotiations last year.
Allies of Ms Mordaunt said she had "nothing but respect" for Lord Frost and would "always fight for Brexit".
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 upcoming TV 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."
Ms Truss's team said Thursday's result showed the foreign secretary was "attracting a wide range of supporters from across the Conservative Party".
A spokesperson for Ms Truss said: "Now is the time for MPs to unite behind the candidate who will cut taxes, deliver the real economic change we need, continue to deliver the benefits of Brexit and ensure Putin loses in Ukraine".
Launching her campaign earlier, Mr Truss pledged to boost the economy through tax cuts and said she was "ready to be prime minister from day one".
Earlier, 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.
Culture Secretary Nadine Dorries, a staunch ally of Mr Johnson who is backing Ms Truss, said: "Now is the time for us all to unite behind a candidate who actually has the ability to lead the country as PM."
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".
Conservative leadership timetable
Monday 18 July - date for third round of voting, with result at 20:00
Tuesday 19 July - date of fourth round of voting, with result at 15:00
Wednesday 20 July - date of fifth round of voting, with result at 16:00
Thursday 21 July - deadline for deciding final two candidates
Monday 5 September - winner announced
Ms Braverman told the BBC she was "disappointed" with the result but was "blown away by the work of her team" and the support she received.
She had styled herself as the only "authentic" Brexiteer candidate in contention, pointing to the fact that she resigned as a Brexit minister under former Prime Minister Theresa May over her deal.
The former barrister has served as MP for Fareham in Hampshire since 2015, replacing Geoffrey Cox as attorney general in 2020.
Meanwhile, on a trip to Germany, Labour leader Sir Keir Starmer said he wanted a general election to provide a "fresh start" for the UK as Mr Johnson steps down.
Sir Keir said: "I don't think the answer to the stagnation of the last 12 years is to change the candidate or the leader of the Tory Party."
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