Boris Johnson responded to allegations of a conflict of interest in an interview saying he acted with "complete propriety".
The Sunday Times reported that Ms Arcuri, an American who moved to London seven years ago, was given £126,000 in public money and was treated to privileged access to three foreign trade missions led by Mr Johnson.
Mr Johnson denied allegations of conflict of interest over the funding, and insisted he acted with "complete propriety" over allegations of a conflict of interest.
In an interview with Sky News in New York, he said: "Everything was done with complete propriety. Completely in the normal way."
He added: "All I can say is I'm very proud of what we did when I was mayor of London, very proud of everything that we did, particularly banging the drum for our city and country around the world."
The PM declined to clarify the nature of his relationship with the woman.
When Boris Johnson was asked to further explain, he replied: "If you will forgive me we are going to talk exclusively about what we are doing here in the UN."
He said: "I'm here to talk about what we are doing in the UN and this country's commitment to tackling climate change stopping the loss of biodiversity and our role as serving as a bridge between our European friends..."
Ms Arcuri, 34, did not comment to the Sunday Times about the nature of her relationship with the PM.
"Any grants received by my companies and any trade mission I joined were purely in respect of my role as a legitimate businesswoman," she said in a statement to the paper.
Shadow minister for the Cabinet Office Jon Trickett has called for the PM to fully explain the matter.
"Boris Johnson must now give a full account of his actions in response to these grave and most serious allegations of the misuse of public money in his former role as mayor of London," the Labour MP said.
"The public has a right to know how and why these funds were used for the benefit of a close personal friend without on the face of it legitimate reason."
In the interview, Boris Johnson twice declined ruling out suspending Parliament for a second time.
He said: "Yes this an issue that has split my party, it split the country, it's a very divisive issue. My strong view is the best way, the only way forward now is to do what the people mandated us to do and come out of the EU on October 31."
The PM also twice declined to rule out resignation if the Supreme Court finds against the Government.
"I'm going to wait and see what the judgment is," he said.