Scotland Yard is assessing two allegations of electoral fraud after claims the Tories offered peerages to senior Brexit Party figures to persuade them to stand down in the General Election.
Lord Falconer wrote to the Metropolitan Police Commissioner and Director of Public Prosecutions calling for a probe into what he said were "exceptionally serious allegations".
The Labour former lord chancellor said it should be investigated as a matter of urgency and must be looked at by police in order to maintain public confidence in the integrity of the election.
His letter to Dame Cressida Dick and Max Hill QC refers to Nigel Farage's claim that he and eight other senior figures within the Brexit Party were offered peerages.
Lord Falconer said he has seen evidence that the claims are correct, telling BBC Radio 4's Today Programme: "I've seen evidence from Ann Widdecombe that is a film in which first she says first of all that somebody from Number 10, not Boris Johnson, not Sir Edward Lister, not Dominic Cummings, first of all pushed her not to stand with, quote, 'moral arguments', then came back and pushed her not to stand on the basis that she be given a place in the negotiating team.
"The law is that if somebody corruptly induces or procures another person to withdraw from being a candidate at an election, that is both a crime and a corrupt practice at an election, which can lead an election to be set aside."
The Metropolitan Police said: "The MPS has received two allegations of electoral fraud and malpractice in relation to the 2019 General Election.
"The MPS special enquiry team is responsible for investigating all such criminal allegations. Both allegations are currently being assessed.
"The MPS will not be providing comment about individual cases."
Mr Farage has claimed he had repeatedly been offered a seat in the House of Lords in an attempt to persuade him to "go quietly".
He said that when that failed, people working "deep inside Number 10" had tried to bypass him, going directly to senior Brexit Party figures and suggesting eight of them could be made peers if they could persuade him to withdraw more of his candidates.
In his letter, Lord Falconer cites the law which refers to "bribery" and "corruptly", inducing or procuring someone to withdraw from being a candidate at an election.
Commenting on the letter on the Today Programme, Michael Gove said: "I've got great respect for Charlie Falconer (Lord Falconer) but I think that this sounds pretty nonsensical to me."
Lord Falconer said that as breaches of the 1983 Act may have taken place, he is "formally requesting that the Director of Public Prosecutions do institute the necessary investigations and commence such prosecutions as he sees fit".
The Prime Minister acknowledged that there may have been "conversations" between senior Tories and people in the Brexit Party, but flatly denied there had been any offers of peerages, saying that was "just not the way we operate".
Brexit Party MEP Ben Habib, when asked if the Tory Party's denials were a lie, told Sophy Ridge on Sky: "Well I'm saying they're being economical with the truth for sure. Absolutely. It has been going on. Without a shadow of a doubt."
Lord Falconer said: "These are exceptionally serious allegations which the DPP must, in accordance with his statutory duty, fully investigate as a matter of urgency.
"In addition, in order to maintain public confidence in the integrity of our electoral processes and this election, it is crucial that the Metropolitan Police also examine these accusations."
The row came amid growing pressure on Mr Farage in the run-up to the close of nominations on Thursday to stand down Brexit Party candidates in all but a few dozen constituencies to avoid splitting the pro-Leave vote.
The Brexit Party leader had already said they would not contest the 317 seats which the Conservatives had won in the 2017 election.
Suspicions that individual Brexit Party candidates were coming under pressure to stand aside were heightened after the prospective candidate for Dudley North announced he would not be running.
Rupert Lowe, a Brexit Party MEP and former chairman of Southampton FC, revealed his decision as nominations were closing - meaning it was too late for the party to put forward an alternative.
Lord Falconer's letter includes a mention of Ann Widdecombe, the former Tory MP now standing for the Brexit Party in Plymouth Sutton and Devonport.
Ms Widdecombe said she was offered a role in the Brexit negotiations if she was prepared to stand aside.
Pressed on Ms Widdecombe's claim, Mr Gove said: "I haven't spoken to Ann and I don't know anyone who has spoken to Ann for months now."
Liberal Democrat leader Jo Swinson told the PA news agency: "Nothing would surprise me about the Conservatives these days given what they've been prepared to do.
"If Boris Johnson's prepared to lie to the Queen, lie to the country, you know, I'm going to stop being shocked at where his lack of boundaries lies."