The prime minister faces allegations he failed to declare a potential conflict of interest over a personal friendship with Jennifer Arcuri, a US technology entrepreneur, who was given a total of £126,000 in taxpayer money.
However Mr Johnson told ITV News: "I can tell you everything was done with full propriety and in accordance with proper procedures."
The Sunday Times reported that Ms Arcuri, an American who moved to London seven years ago, was treated to privileged access to three foreign trade missions led by Mr Johnson.
The US entrepreneur received a £100,000 grant from the Department of Digital, Culture, Media and Sport earlier this year.
In response to whether he was involved in the award of the money, Mr Johnson said: "Of course not."
He added: "I am very, very happy with what we did in the London mayolty to promote our city and country."
The prime minister declined to clarify the nature of his relationship with Ms Arcuri.
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.
Mr Johnson had earlier repeatedly refused to comment on the allegations centring on his association with the American businesswoman, during sustained questioning on his flight to New York for the United Nations General Assembly.
He did not directly respond to six questions, including whether the allegations could be solved with a public enquiry.
Shadow minister for the Cabinet Office Jon Trickett has called for the prime minister 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."