Conservative MPs will vote again on Tuesday to select the two candidates for leader who will be put to a vote of their peers.
When the results are announced later, one of the four remaining contenders – Rishi Sunak, Penny Mordaunt, Liz Truss, and Kemi Badenoch – will be eliminated.
Mr Sunak appears to be one of the final two candidates, with the other three vying for second place.
Tom Tugendhat, a senior backbencher, was defeated in a vote of MPs on Monday.
Separately, Prime Minister Boris Johnson will hold his final scheduled cabinet meeting before the summer recess in Westminster on Tuesday.
The Conservative Party wants MPs to have chosen the two candidates to replace him before the Commons finishes on Thursday – with the final vote set to take place on Wednesday.
After a summer of campaigning by the two final contenders, grassroots Tories will vote for who should become leader, with the winner to be announced on 5 September.
Mr Tugendhat was eliminated after receiving 31 votes, down one from his performance in the second round last Thursday.
The other candidates placed in the same order as on Thursday, with Mr Sunak coming out on top having gained 14 votes for a total of 115.
Ms Mordaunt dropped one vote but held onto second place with 82 votes, while Ms Truss gained seven votes to bring her total up to 71.
Ms Badenoch placed fourth with 58 votes, up nine from the previous round.
In a statement, Mr Tugendhat said: “I want to thank my team, colleagues and, most of all, the British people for their support.
“I have been overwhelmed by the response we have received across the country. People are ready for a clean start and our party must deliver on it and put trust back into politics.”
But Ms Mordaunt said: “My vote is steady and I’m grateful to my colleagues for all their support and thrilled to be in second place once more.”
A prominent Sunak supporter said: “That’s a cracking result – Rishi gaining most [votes].”
On Tuesday morning, a source close to Ms Badenoch’s campaign said she was the candidate offering real change. “Mordaunt has gone backwards and Truss only added seven votes. Kemi has the greater momentum and it’s all to play for today with a group of colleagues who want change.”
In the end Liz has had three attempts at getting to the final two and hasn’t made a breakthrough.”
A source within Ms Truss’s campaign told the BBC on Tuesday: “We’re confident we’ll make it through. MPs who don’t want Rishi Sunak to be PM need to think about who is most likely to beat him in the final two, and can turn the economy round and win an election. Liz is the best bet.”
A Sky News debate scheduled for Tuesday was cancelled after Mr Sunak and Ms Truss declined to take part, and amid concern among senior Tories that airing disagreements in public could damage the party.
Meanwhile, Mr Johnson is expected to praise the resilience of schools and the NHS in continuing to function during the current heatwave when his cabinet meets formally for the last time on Tuesday.He will also use the occasion to mark a year since England moved to step four of the so-called Covid roadmap, which saw the end of most of the remaining lockdown restrictions.Monday also saw the government win a vote of confidence in itself in the Commons by 349 votes to 238, a majority of 111.The vote could have triggered a general election had the government lost, although that would have required dozens of Conservative MPs to side with Labour and other opposition parties.
