Penny Mordaunt and Liz Truss are locked in a tight race for a chance to become prime minister, alongside Rishi Sunak, as polling seen by the Guardian suggests Labour could beat any of the three at a general election.

The former levelling up minister Kemi Badenoch was eliminated in Tuesday’s ballot of Conservative MPs, coming fourth among the remaining contenders with 59 votes.

Sunak remained in the lead with 118 votes, just short of the 120 needed to secure a spot in the last two. Mordaunt, the international trade minister, kept her second spot with 92 votes, up from 82, but Truss, the foreign secretary, closed the gap by picking up 15 extra votes to reach 86.

The three contenders will be whittled down to two on Wednesday afternoon, and Tory party members’ final decision on who will become prime minister will be announced on 5 September.

Polling by Electoral Calculus and Find Out Now asked the public in the past few days how they would be likely to vote if Sunak, Truss or Mordaunt were Conservative leader.

The results point to a nine percentage point lead for Labour over a Mordaunt-led Tory party, or 12 points over Sunak or Truss. That would give Keir Starmer, the Labour leader, a slim outright majority of two seats over Mordaunt or a more workable 24 seats over the other two.

Martin Baxter, the chief executive of Electoral Calculus, said: “The public are not very keen on voting for a Conservative party led by any of the leading candidates, which opens the way to Downing Street for Keir Starmer at the next election.”

Labour put out an attack video on Tuesday splicing together critical comments the candidates have made in televised debates, including Truss saying to Sunak: “Under your plans, we are predicted to have a recession.”

Truss’s gains in Tuesday’s ballot came as a surprise to some backers of Tom Tugendhat, who dropped out on Monday, freeing up his 31 supporters to go elsewhere. Tugendhat is on the centrist wing of the party and few of his supporters had been expected to swing behind the rightwinger Truss.

Kemi Badenoch.

While Mordaunt was up 10 votes, Truss’s team will hope to win over many of Badenoch’s supporters, potentially allowing her to move into second place in Wednesday’s ballot.

One former Tugendhat supporter said: “I suspect all the campaigns are going to be very tense tonight: you can easily construct a scenario where they are all on a very similar level of support tomorrow.”

A Sunak-supporting MP denied the former chancellor’s camp would be disappointed at not having reached the 120-vote threshold on Tuesday to guarantee a place among the final two. “Every ballot we’ve gone in the right direction, so we need to make those arguments now with the 59 colleagues who voted for Kemi,” they said.

(Left to right) Rishi Sunak, Liz Truss and Penny Mordaunt.

Steve Baker, a strong supporter of Badenoch, said he believed the MPs who had voted for her would want to “take stock for a couple of hours” before deciding who to back in Wednesday’s fifth and final voting round.

“But I would have thought that most of the people who are attracted to Kemi, they’ll mostly not be attracted to Penny,” he said. “So obviously I’m hopeful that Liz will be able to attract their support. But one must never take anything for granted.”

Baker said he did not believe any campaign was trying to skew the result by lending out blocks of votes as a way of eliminating other candidates, while stressing that MPs were “a sophisticated electorate” and some could vote tactically on an individual basis.

Badenoch, who has never held a cabinet post, styled herself as a straight-talking reformer and won the support of Michael Gove, the levelling up secretary who was sacked by Johnson as he fought to save his collapsing government.

By making it to the final four she is likely to have secured a senior role in the administration of the new prime minister, whoever wins.

The past 24 hours have brought a flurry of policy announcements from the remaining candidates, with Mordaunt promising to scrap housing targets and press ahead with Johnson’s project of levelling up, and Truss pledging to boost defence spending to 3% of GDP.

Sunak, meanwhile, claimed he was the only candidate who could save the UK, promising to rule out a Scottish independence referendum and attack Labour over claims it could form a coalition with the Scottish National party.

Sir Graham Brady, the chair of the backbench 1922 Committee who announced the results of the ballot, said one Conservative MP had spoiled their paper and another had not voted.

The Bournemouth East MP, Tobias Ellwood, was prevented from taking part after Johnson removed the Tory whip from him as punishment for failing to vote on Monday evening’s motion of no confidence.

Find Out Now polled a weighted sample of 1,261 adults online on 18-19 July about how they would vote in an immediate general election if each Tory candidate were leader.