Boris Johnson has repeatedly refused to answer questions over allegations that he granted thousands of pounds of public money to "close friend" Jessica Arcuri.
It is claimed the prime minister visited the tech entrepreneur's flat "several times" while he was serving as the mayor of London.
In an interview with Sky News, Boris Johnson dodged questions over a report in The Sunday Times which revealed that Ms Arcuri was given a total of £126,000 in public money and privileged access to three overseas trade missions led by the prime minister.
Who is Jennifer Arcuri?
She is an American model-turned-businesswoman who reportedly described Mr Johnson as "one of her best friends".
The 34-year-old moved to London seven years ago but has since moved back to the US, and now lives in California.
According to her profile on IMDb, Ms Arcuri began her career as Radio Disney DJ Razzle Dazz at the Walt Disney Company.
She then tried her hand at movies - producing and directing a short film called LA Valise in 2008, which became part of the Short Film Showcase at the Cannes Film Festival.
But in 2011, she moved to the UK where she launched a career in tech and founded Playbox Ltd - a social network for the video tech industry.
She was living in a top-floor flat in Shoreditch, east London - and according to The Sunday Times, it featured a dancing pole in the living room.
On her Twitter profile, which features a cover photo of Tower Bridge in London, Ms Arcuri describes herself as an entrepreneur, cybersecurity expert and producer.
She is also the founder of Innotech Network, which claims to join investors, entrepreneurs and policymakers in "stimulating the London tech scene".
Her relationship with Boris Johnson
Ms Arcuri joined the Conservative campaign team in 2012 as the prime minister was seeking re-election for a second term as mayor.
It is alleged in The Sunday Times Ms Arcuri was given preferential treatment to ensure she could come on trade delegations, with Mr Johnson's team ensuring she would accompany him on trips even though her initial applications to take part were refused.
Sources claimed in the newspaper that Mr Johnson was a regular visitor to her Shoreditch flat in east London - and that in 2013 her fledgling company received £10,000 in sponsorship from an organisation that Mr Johnson was responsible for as London mayor.
The owner of her flat's building reportedly told The Sunday Times that she referred to the prime minister as "one of her best friends".
Earlier this year, Hacker House, a company she is registered as a director of, won a £100,000 grant from the Department for Digital, Culture, Media and Sport from a fund for English-based entrepreneurs, even though she now lives in the US.
Whitehall officials are now investigating the grant.
Her husband Matthew Hickey claimed on Twitter Ms Arcuri was nothing to do with it, and said they had not received any money, adding "so far all I've seen is a press release".
Labour has demanded a full explanation for the use of funds, suggesting it was a "misuse of public money".
What has Boris Johnson said?
Stopping short of directly addressing accusations relating to his relationship with Ms Arcuri, the prime minister told Sky News: "We did a huge amount of work while I was mayor of London selling our fantastic city around the world, beating the drum for London and the UK.
"I'm very proud of what we did. Everything was done with complete propriety and in the normal way."
During a flight to New York, Mr Johnson refused to answer six times when asked about the issue by reporters.
He was asked: "You're accused on the front page of The Sunday Times today for failing to declare a series of conflicts of interest over your friendship with Jennifer Arcuri.
"This is your first opportunity to reply to that story. Would you like to deny the story?"
He replied: "I'm here to talk about what we're doing in the UN, and this country's commitment to tackling climate change, stopping the loss of biodiversity and our role as a bridge between our European friends and Americans when it comes to the crisis in the Gulf and many other issues."
Ms Arcuri told The Sunday Times: "Any grants received by my companies and any trade mission I joined were purely in respect of my role as a legitimate businesswoman."
The newspaper added she did not comment on the nature of her relationship with Mr Johnson.