Charlotte Edwardes said the incident took place at the offices of The Spectator magazine in London shortly after Mr Johnson became editor in 1999.
She accused the prime minister of squeezing her thigh, saying he "had enough inner flesh beneath his fingers to make me sit suddenly upright".
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."
Downing Street has denied the claim, with a Number 10 spokesman saying: “This allegation is untrue.”
Ms Edwardes responded to the denial on Twitter, saying "if the prime minister doesn’t recollect the incident then clearly I have a better memory than he does".
Health Secretary Matt Hancock, who had earlier sought to play down the allegations, told Channel 4 News he knows and trusts the journalist.
He told Krishnan Guru-Murthy: "I know Charlotte well and I entirely trust what she has to say."
He added: "I know her and I know her to be trustworthy."
Ex-minister Amber Rudd echoed the health secretary, writing on Twitter: "I agree with Matt Hancock."
Ms Edwardes made the allegations in her first column for The Sunday Times, where she wrote: "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."
Labour's 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.
Mr Hancock's original comments played down the report, saying there were "always lots of other stories in papers".
Speaking at a Tory party fringe event in Manchester hosted by HuffPostUK, he said: "Boris has never lectured other people about their private lives.
"I think that we should concentrate on delivering on what we are in politics for, which in my view is to serve the citizens of this country."
Ms Butler said Mr Hancock's remarks were "dismissing sexual harassment allegations as a private matter".
She called on him to apologise "for his disgraceful remarks".