The allegations, which Downing Street has so far declined to comment on, involve funds of more than £120,000 given to technology entrepreneur named Jennifer Arcuri during Boris Johnson's time as Mayor of London, according to The Sunday Times.
For her part, Ms Arcuri said: "Any grants received by my companies and any trade mission I joined were purely in respect of my role as a legitimate businesswoman."
She did not comment on the nature of her relationship with Mr Johnson.
What are the allegations?
According to the newspaper report, Ms Arcuri was given a total of £126,000 of public money for her newly-launched tech businesses.
She reportedly first received £10,000 in sponsorship from an organisation overseen by Mr Johnson when he was in City Hall in 2013.
A year later, her business received a £15,000 government grant under the Sirius scheme, which supports visas for young entrepreneurs to build their businesses in Britain.
And then earlier this year her newest business was given a £100,000 grant from the Department of Digital, Culture, Media and Sport. This award is being investigated, however, as it is thought that Ms Arcuri had already moved back to the US when she received the money.
She was also given privileged access to three foreign trade missions led by Mr Johnson while he was mayor, despite the fact her businesses had not met the eligibility criteria.
According to an honesty clause in the Greater London Authority's code of conduct, Mr Johnson had a responsibility to declare private interests relating to public duties. He was also banned from providing any undue benefits to friends.
READ MORE: Boris Johnson declines to deny conflict of interest over public money funding of model friend
Who is Jennifer Arcuri?
Jennifer Arcuri, 34, is a model-turned-technology entrepreneur. She is American but moved to London seven years ago. She has since moved back to the States.
A profile on film database site IMDb says Ms Arcuri began her career as Radio Disney DJ Razzle Dazz at the Walt Disney Company.
She later moved into cinema and produced and directed short film La Valise in 2008, which was part of the Short Film Showcase at the Cannes Film Festival.
Ms Arcuri moved to the UK in 2011 where she launched a career in tech. She founded Playbox Ltd, a social network for the video tech industry.
She also founded The Innotech Network, which sought to join together investors, entrepreneurs and policymakers around Silicon Roundabout in east London. Mr Johnson is believed to have delivered a keynote speech at Innotech's first summit in 2012, when Ms Arcuri was 27. He went on to appear at subsequent events.
According to her Twitter profile, Ms Arcuri is also the founder of Hacker House and describes herself as an entrepreneur, cyber security expert and producer.
What is her relationship with Johnson?
It has been reported the pair are close friends.
According to the newspaper, Mr Johnson regularly visited Ms Arcuri at her flat in Shoreditch, east London, when he was Mayor of London between 2008 and 2016. The owner of the building told The Sunday Times she referred to him as "one of her best friends".
Her home reportedly featured a dancing pole in the living room.
She campaigned for Mr Johnson to be mayor for a second term and posted photos of him on social media.
Mr Johnson has refused to comment on suggestions that he had sexual relations with Ms Arcuri.