The levelling-up secretary, who is divorced from columnist Sarah Vine, went on to have a relationship with the female friend of the business secretary, according to The Times. The woman has since separated from her husband.
The relationship between Mr Gove and the woman is believed to have come to an end, but a source told the newspaper it caused a “significant deterioration” between the two cabinet colleagues.
It is understood that it had such an impact that the former close political allies were not on speaking terms earlier this year.
They are said to be talking again now and have re-established a working relationship, but are no longer close.
A source told The Times: “They had a significant falling out over this – there was a significant deterioration in relations between them.”
Last year, Mr Gove backed Ms Badenoch in the Tory leadership contest to replace Boris Johnson, describing her as “brave, principled, brilliant and kind”.
Giving her his full support, he said at the time: “Kemi doesn’t just win the argument, she delivers – on getting the Whitehall machine to embark on new policies and on levelling up Britain.
“Now she has the opportunity to use her first-class brain to fix the big problems facing our country.”
He went on to say she would be “Sir Keir Starmer’s worst nightmare”.
Ms Badenoch finished fourth in the contest behind Liz Truss, Rishi Sunak and Penny Mordaunt, but is considered a potential future leader of the party.
Nadine Dorries this week claimed Mr Gove had been mentoring Ms Badenoch “for a long time” but she was now “distancing herself from Gove”.
In her book The Plot: The Political Assassination of Boris Johnson, the former culture secretary claimed Mr Gove had been “building up Kemi Badenoch as the next leader of the Conservative Party”.
But Ms Dorries wrote: “She is distancing herself from Gove, whether she is being advised to – and I use that word reservedly – or she has realised the dangers of being seen to be too close to him. She’s second only to Ben Wallace in the opinion polls.
“The trajectory Kemi is on, she must think to herself, I don’t need Gove any more.”
Ms Dorries criticised Ms Badenoch on Saturday, highlighting a message she sent during the downfall of Mr Johnson’s administration last summer, urging a fellow member of the government to resign. Leaked WhatsApp messages revealed she told assistant government whip Sarah Dines: “Resign before midnight. DO IT. DO IT. DO IT.”
Resharing the post on X, formerly Twitter, Ms Dorries claimed it showed “the real Kemi Badenoch”.
She also claimed to have had a bet with several friends that the publication of The Plot would lead to stories emerging of Ms Badenoch distancing herself from Mr Gove. “It would appear that I’ve won,” she said.
Mr Gove finalised his divorce from Ms Vine last year. The Daily Mail columnist said her husband’s “only mistress was politics” as she admitted she had worked to prevent him becoming prime minister during their 20-year marriage.
In an interview with Tatler magazine, she said: “The mistress wins in the end. It’s all about making sure she is happy.
“In my case, despite all the rumours, his only mistress was politics. That’s what he is in love with. He is genuinely in love with politics.”
A spokesperson for Mr Gove declined to comment. Ms Badenoch was approached for comment.
News of the affair came as The Mirror reported that Labour deputy leader Angela Rayner had split from her partner, fellow Labour MP Sam Tarry.