Boris Johnson has denied any impropriety following claims he failed to declare a potential conflict of interest while London mayor.
The Sunday Times said Jennifer Arcuri - who knew Mr Johnson - joined trade missions he led and received thousands of pounds in sponsorship grants.
The PM earlier refused address the allegations, but later said "everything was done entirely in the proper way."
She told the paper it was part of her role as a legitimate businesswoman.
Labour has said Mr Johnson must give a full account of his actions, but pressed by journalists during a flight to New York on Sunday night, the now-prime minister refused to comment.
On Monday evening, though, he told the BBC's John Pienaar: "All I can say is I am very proud of what we did as Mayor of London... particularly banging the drum for our city and country around the world."
He added: "I can tell you that absolutely everything was done entirely in the proper way."
Technology entrepreneur Ms Arcuri is believed to have moved to London seven years ago - Mr Johnson was mayor between 2008 and 2016.
She joined a joined a number of trade missions led by him while in office, and it is understood she attended events on two of these trips - to New York and Tel Aviv - despite not officially qualifying for them as a delegate.
The Sunday Times reported that one of her businesses received £10,000 and £1,500 in sponsorship money from a mayoral organisation when Mr Johnson was mayor, as well as a £15,000 government grant for foreign entrepreneurs to build businesses in Britain.
The newspaper also said Ms Arcuri got a £100,000 grant from the Department of Culture, Media and Sport earlier this year.
The grant was intended for "English-based" businesses - although she had moved back to the US in June 2018.
The Sunday Times said it had found the registered address on the grant application form was a rented house in the UK and no longer connected to her.
The government has confirmed to the BBC it is investigating, but said the funds were awarded to a UK-registered company.
Transport Secretary Grant Shapps said it was "perfectly normal" for entrepreneurs to join trade missions, aimed at promoting British businesses overseas.
He told the BBC: "British companies and entrepreneurs go on trade missions. It's quite right and proper and I'm sure that's exactly what's happened there."
Government Minister @grantshapps says "trade missions take place all the time" and he was surprised to see a story about Boris Johnson's alleged misuse of public finds while mayor in The Sunday Times #r4today https://t.co/IuCitVtzxj pic.twitter.com/b4mnuhNBN3— BBC Radio 4 Today (@BBCr4today) September 23, 2019
The current London Mayor, Labour's Sadiq Khan, said he had ordered City Hall officials to look into the allegations.
Journalists asked Mr Johnson about the allegations when travelling with him to the UN General Assembly in New York.
The PM told reporters he was there to "talk about what we're doing in the UN and this country's commitment to tackle climate change", as well as "the crisis in the Gulf and any other issues that may arise".
Asked again, he replied: "I'm here to talk exclusively about the work of the UN."
Ms Arcuri was quoted by the Sunday Times as saying: "Any grants received by my companies and any trade mission I joined were purely in respect of my role as a legitimate businesswoman."
Who is Jennifer Arcuri?
The woman at the centre of this story is Jennifer Arcuri, who describes herself on Twitter as an entrepreneur, cyber security expert and producer.
She began her career as a DJ on Radio Disney, before moving into film - where she wrote, produced and directed a short film that went on to be sold at Cannes Film Festival.
Ms Arcuri then brought in her tech skills to create a streaming platform for independent film makers.
But it was her founding of The Innotech Network in London that saw her path cross with Boris Johnson.
The network hosts events to discuss tech policy, and Mr Johnson was the keynote speaker at the first of those in 2012.
Since then, Ms Arcuri has also founded another company called Hacker House, which uses ethical hackers to find tech solutions for businesses.