Kate Maltby, who is three decades younger than the First Secretary of State, told The Times that Mr Green had “fleetingly” touched her knee during a meeting in a Waterloo pub in 2015 and a year later sent her a “suggestive” text message after she was pictured wearing a corset in the newspaper.
Mr Green said any allegation that he made sexual advances to Ms Maltby was “untrue (and) deeply hurtful”.
He is the most senior politician yet to be caught up in a tide of allegations and rumours relating to sexual harassment and abuse swirling around Westminster.
Labour has launched an independent inquiry into claims that prominent activist Bex Bailey was discouraged by a party official from reporting an alleged rape at a Labour event in 2011 on the grounds it might damage her political career.
And in a separate case, a woman who claims she was sexually assaulted by an MP on a foreign work trip last year has said her allegations were not taken seriously.
Ms Maltby, 31, said that 61-year-old Mr Green was an old friend of her parents who she had approached for advice after becoming involved in Tory activism.
When they met for drinks, she said he suggested could help her start a political career, before turning the conversation to the subject of affairs at Westminster.
Ms Maltby said that he mentioned that his own wife was “very understanding” and she then “felt a fleeting hand against my knee - so brief it was almost deniable”.
Angered by the incident, she had no further contact with Mr Green until his text a year later saying he had “admired you in a corset” and inviting her for a drink.
Writing in The Times, she said she renewed contact with Mr Green after his appointment to the Cabinet, but doubted he knew how “awkward, embarrassed and professionally compromised” she felt about the alleged incident.
Mr Green said it was “absolutely and completely untrue that I’ve ever made any sexual advances on Ms Maltby”.
His text was sent in the spirit of “two friends agreeing to meet up for a regular catch-up”, he said, adding: “This untrue allegation has come as a complete shock and is deeply hurtful, especially from someone I considered a personal friend.”
A Downing Street spokesman said that Mrs May had asked Cabinet Secretary Sir Jeremy Heywood to “establish the facts and report back as soon as possible”.
Meanwhile, Labour leader Jeremy Corbyn vowed he would allow “no tolerance” of sexism, harassment or abuse after Ms Bailey spoke out about the party’s failure to support her following her alleged rape.
Ms Bailey, 25, a former member of the party’s National Executive Committee, said her attacker was not an MP, but someone more senior than her in the party.
Aged 19 at the time of the attack, she said she felt too scared and ashamed to report it to the police, but eventually summoned up the courage to tell a senior party official.
Labour said it takes the allegations “extremely seriously” and has launched an independent inquiry by general secretary Iain McNicol into the claims that she was not given adequate support by the party.
In a another development, ITV News carried an interview with a woman, whose identity was hidden, who said she was pushed onto a hotel bed by an unnamed MP during a 2016 overseas trip.
Despite her making clear she did not welcome the advance, the man was “insistent” and tried to kiss her as he held her down, she said.
Lawyers for the MP concerned told the broadcaster he categorically denied the claims.