Boris Johnson is under fire for failing to provide details of his contacts with US businesswoman Jennifer Arcuri to a London Assembly inquiry.
The PM responded to the assembly's request for information on Tuesday.
But the assembly said the letter it received - marked "confidential and not for publication" - did not answer any of its questions.
Mr Johnson denies claims of a conflict of interest over his friendship with Ms Arcuri when he was London mayor.
The assembly had asked for details and a timeline of all contact between the pair, including private text messages and emails.
A London Assembly spokeswoman told the BBC the letter "doesn't answer any of the questions we asked", adding: "I can't understand why it is labelled confidential."
The assembly is now seeking legal advice over whether members of its oversight committee can discuss the contents of the letter at their meeting next week.
In a statement, Len Duvall, Labour chairman of the committee, said: "We did finally receive a response from Boris Johnson, through his solicitors, which they have indicated may not be published. At this stage we are respecting that, but we are seeking further clarification.
"Nothing in the response, in our opinion, reflects the need for confidentiality. In fact, the response is insufficient as far as our request for information is concerned.
"We are focused on our investigation and considering next steps. A number of options are open to us; they include speaking to various people and using our power of summons."
He said the committee was liaising with the Independent Office for Police Conduct, which has been asked to consider whether Mr Johnson, who as mayor was responsible for policing in London, should be investigated for misconduct in public office.
Labour's shadow Cabinet Office minister Jon Trickett said: "With an issue as serious as potential abuse of public office, it is absolutely in the public interest that this letter be published.
"Boris Johnson might think he is above the law but he cannot hide from scrutiny."
If the PM fails to answer the assembly's questions, added Mr Trickett, "he is showing contempt for the inquiry and the people of this country."
Mr Johnson held the office of London mayor between 2008 and 2016.
According to the Sunday Times, which first reported the story, technology entrepreneur Ms Arcuri joined trade missions led by Mr Johnson when he was mayor and received thousands of pounds in public money.
It is also understood she attended events on two of the trade missions - to New York and Tel Aviv - despite not officially qualifying for them as a delegate.
Ms Arcuri told ITV's Good Morning Britain Mr Johnson was "a really good friend" who had spoken at event she organised - but denied the then mayor had shown any "favouritism" towards her.
The code governing conduct at London City Hall states that public office holders should not act in any way to gain benefits for families or friends, and should declare private interests to resolve any conflicts.
The prime minister has denied breaking any rules of conduct and insisted everything was done "entirely in the proper way".
Separately, the current Mayor Sadiq Khan has asked a senior lawyer to review a 2013 decision by London and Partners, the mayor's promotional agency, to sponsor a conference organised one of Ms Arcuri's companies, for £10,000.
London and Partners say they have found no evidence of Mr Johnson's involvement in the decision.
The Department for Culture, Media and Sport is, meanwhile, "reviewing" a £100,000 grant made in February this year to Ms Arcuri's cyber-security business Hacker House.