Downing Street was forced to deny claims that Mr Johnson squeezed the thigh of a female journalist under the table during a private lunch.
Charlotte Edwardes said the incident took place at the offices of The Spectator in London shortly after Mr Johnson became the magazine's editor in 1999.
After the lunch, she said she had confided in the young woman who was sitting on the other side of Mr Johnson, who told her: "Oh God, he did exactly the same to me."
A Number 10 spokesman said: "This allegation is untrue."
But Health Secretary Matt Hancock said he believed Ms Edwardes was trustworthy.
The Chancellor tried to evade questions about the row in a series of broadcast interviews.
"I don't think it's a good idea to get drawn into personal allegations," he told the BBC.
"For my part, I'm not going to get into that.
"The Prime Minister has said that this is completely untrue. I have full faith in the Prime Minister, I don't doubt what he has said for a second but I'm not going to get drawn into these allegations."
Separately, the Prime Minister had to insist there was "no interest to declare" amid a storm over his association with the American businesswoman Jennifer Arcuri when he was mayor of London.
In a sign of Tory frustration that questions about Mr Johnson's behaviour are taking attention away from policy announcements at the Conservative Party conference, Mr Javid acknowledged he would "rather be talking about the infrastructure revolution" he has promised.
Former Tory cabinet minister Justine Greening, who now sits as an independent, told BBC Radio 4's Today programme: "I can't comment on those accusations but they are deeply concerning and in a sense they go to the heart of this question about character and integrity of people in public life and what standards the electorate have a right to expect."
The claims about Mr Johnson's behaviour were made by Ms Edwardes in her first column for The Sunday Times.
She said: "I'm seated on Johnson's right; on his left is a young woman I know.
"More wine is poured; more wine is drunk. Under the table I feel Johnson's hand on my thigh. He gives it a squeeze.
"His hand is high up my leg and he has enough inner flesh beneath his fingers to make me sit suddenly upright."
Shadow secretary for women and equalities Dawn Butler said it was a "shocking but sadly all too familiar story".
"What is it about powerful men feeling entitled to harass women? Boris Johnson has serious questions to answer," she tweeted.