Boris Johnson has said claims that Brexit Party election candidates have been offered peerages to encourage them to stand down are “nonsense”.
Leader Nigel Farage has said senior figures have been offered inducements, such as government jobs, in return for not running against Conservatives.
Ann Widdecombe, a Brexit Party candidate, said she would swear on the Bible she had been approached.
The PM told the BBC that “certainly no-one’s been offered a peerage”.
Taking part in a question-and-answer session on Radio 5 Live, he responded to Mr Farage’s comments, telling host Rachel Burden: “What is this nonsense? I am sure there are conversations that take place between politicians of all parties but certainly nobody’s been offered a peerage.”
Conservative Party chairman James Cleverly said claims that places in the House of Lords had been dangled in front of their political opponents were “completely unfounded”.
Meanwhile, former lord chancellor and Labour peer Lord Falconer has written to the Metropolitan Police and the Director of Public Prosecutions asking them to investigate Mr Farage’s claims.
Lord Falconer said he believed that the alleged offers could, if proven, amount to a “significant” attempt to influence the election and that authorities should “investigate now”.
The Met Police said it had received two allegations of electoral fraud and malpractice, which are both currently being assessed. It added it would not comment about individual cases.
The Brexit Party is not contesting seats won by the Conservatives at the 2017 general election, but will put up candidates against the party elsewhere, including in many Labour-held marginal constituencies that Mr Johnson hopes to win to secure a parliamentary majority.
Conservatives have urged Mr Farage to reconsider, saying he risks splitting the pro-Brexit vote and allowing Labour – which wants another Brexit referendum – to retain dozens of seats.
Some Brexit Party candidates in marginal Labour seats, such as Rupert Lowe in Dudley North, have decided to withdraw of their own accord.
In a video posted on Twitter on Thursday, Mr Farage said that he, along with eight “senior figures” in his party, had been offered jobs “in the (Brexit) negotiating team and in government departments” while there had been “hints at peerages too”.
He said the offer had been made by people “deep inside Number 10 Downing Street” – although he did not think Mr Johnson was involved.
Ms Widdecombe, a former Tory minister who is standing for the Brexit Party in Plymouth Sutton and Devonport, said she had initially been approached by “someone at Downing Street” who told her she had a “moral obligation” to stand down.
In a subsequent conversation, the MEP said, she had been offered “a role” in the next phase of Brexit negotiations over the UK’s future relationship with the EU.
“I have no idea what that means,” she told the BBC’s Victoria Derbyshire show. “I immediately said that I had played no role in the Tory party for a large number of years and that I couldn’t now be flattered, buttered up or promised things.”
As a practising Catholic, she said was “prepared to put her hand on the book (Bible) over this”.
A Conservative source has told the BBC the Brexit Party candidate in Peterborough was offered an unpaid role in education in the hope it would convince him to stand aside.
Mike Greene is standing for the party in the Cambridgeshire constituency, which Labour held narrowly at a by-election in June.
Mr Greene’s team said the offer of a role had been made to him, but added that their candidate would definitely be running.
The Brexit Party, founded earlier this year, won 29 seats in May’s European elections, the most of any UK party.