No 10 has insisted Andrea Leadsom did not call for Sir Michael Fallon to be sacked from the cabinet amid reports she complained about his conduct.
Sir Michael, who quit office on Wednesday saying his general conduct fell short of expected standards, has "categorically denied" the allegations.
Mrs Leadsom has led ministers' response to the Westminster misconduct claims.
In a statement to Parliament on Monday, she said Commons procedures for handling complaints about MPs needed to be overhauled as women working in Parliament had a right to feel safe.
Sir Michael became the first minister to resign since allegations of inappropriate behaviour by MPs from different parties first surfaced ten days ago.
The latest newspaper claims involving the veteran Tory date back to between 2010 and 2012 when he and Mrs Leadsom were members of the Treasury Select Committee.
According to the Sun and the Daily Mail, Mrs Leadsom remarked to Sir Michael - who was not a minister at the time but was deputy chair of the Conservatives - that she had cold hands and he allegedly replied: "I know where you can put them to warm them up".
The newspapers claim Sir Michael was forced to quit as defence secretary after Mrs Leadsom complained to Prime Minister Theresa May about the alleged incident.
A source close to Sir Michael said he "categorically denies" the newspapers' claims. Mrs Leadsom has not commented on them.
But No 10 issued a statement, saying: "The Leader of the House did not, and has not, asked the prime minister to consider the position of Sir Michael Fallon when he was defence secretary."
Sir Michael confirmed on Tuesday that he was once rebuked by a journalist, Julia Hartley-Brewer, for putting his hand on her knee during a dinner in 2002, and he apologised at the time.
A day later, he resigned as defence secretary, telling the BBC: "The culture has changed over the years, what might have been acceptable 15, 10 years ago is clearly not acceptable now.
"Parliament now has to look at itself and the prime minister has made very clear that conduct needs to be improved and we need to protect the staff of Westminster against any particular allegations of harassment."
He was replaced on Thursday by Chief Whip Gavin Williamson.
Ms Hartley-Brewer said that if he had gone because of her knee, it would be "the most absurd reason for anyone to have lost their job in the history of the universe".
Labour is also facing new claims of sexual harassment and has suspended an MP whilst it carries out an investigation.
Kelvin Hopkins, who has represented Luton North for 20 years, has been accused of sexually harassing a party activist.
A spokesman said Labour "takes all such complaints extremely seriously and has robust procedures in place".