The Metropolitan Police is considering whether to investigate the funding of the lavish refurb of the Downing Street flat.
Lawyers for the Labour Party have claimed there was a “reasonable suspicion” that Boris Johnson had broken anti-bribery laws.
Solicitors wrote to Scotland Yard last week suggesting the force was “duty-bound” to begin a formal investigation.
The Met confirmed they had received the letter which was being considered by officers from its Central Specialist Crime Command.
Labour's complaint follows the release last month of an exchange of WhatsApp messages between the Prime Minister and Tory donor Lord Brownlow.
They show that Mr Johnson discussed a proposal by the peer for a "Great Exhibition 2.0" at the same time as requesting his help with the £112,000 revamp of his official residence.
Ministerial records show that two months later, Lord Brownlow attended a meeting with the then culture secretary Oliver Dowden to discuss the exhibition plan.
There is no suggestion of wrongdoing by Lord Brownlow.
Labour's lawyers said the PM may have acted improperly by having “linked” the request for funds to cover the refurb with a promise to “promote a project” backed by the benefactor.
In his letter, seen by the Guardian, Gerald Shamash, of the law firm Edwards Duthie Shamash, said that despite inquiries by the Electoral Commission and Mr Johnson's adviser on ministerial interests, Lord Geidt, there were still matters that were "uninvestigated and unconsidered".
"It is respectfully suggested that the known facts and the clear, sensible inferences to which some of those facts give rise create such reasonable suspicion that, were the suspect anyone other than the Prime Minister, the Metropolitan Police would rightly consider itself duty-bound to investigate," he wrote.
"Indeed, if anything, the fact that the suspicion arises in relation to someone in such a high office makes it more, not less, important in the public interest that these matters are investigated."
A Met spokesman said: "A letter was received and acknowledged on Friday February 4. It is being considered by officers from the Met's Central Specialist Crime Command. No investigation has been opened."
In response to the claims in the letter, a No 10 spokesman said: "These allegations are categorically untrue and a clear misrepresentation of the facts.
"Lord Brownlow, separate to his work for the emerging Downing Street Trust, put proposals (for a great exhibition) from the Royal Albert Hall, the national institution and charity, to the Prime Minister.
"This was passed to the lead department, DCMS (Department for Digital, Culture, Media and Sport). It is a matter of public record that no project was taken forward by the Government."
Lord Brownlow is a trustee of the Royal Albert Hall Trust and an ambassador of the hall's 150th anniversary.
