Keir Starmer has been urged to sack Peter Mandelson as the UK’s US ambassador after more details emerged of his close friendship and business dealings with the paedophile financier Jeffrey Epstein.

The prime minister was pressed on whether he knew that Mandelson, when business secretary in 2010, had reportedly allowed Epstein to help broker the sale of a UK government-owned business to a US bank – months after the financier had been released from jail in relation to his conviction for procuring a child for prostitution and soliciting a prostitute.

He said that “full due process was gone through when the appointment was made”, although Mandelson added on Tuesday night that he had not discussed the matter with the prime minister.

Starmer backed Mandelson when pressed on the US ambassador’s future by Kemi Badenoch, the Conservative leader, and Ed Davey, the Lib Dem leader, at prime minister’s questions.

The friendship between the two men has come back under the spotlight after Democrat members of the House oversight committee released Epstein’s 50th “birthday book”, in which Mandelson called him “my best pal”, in a handwritten note.

The US ambassador gave an interview to the Sun’s Harry Cole on Tuesday saying that more “very embarrassing” details of his friendship with Epstein would come out but insisting he had never seen any “wrongdoing”.

Epstein killed himself while awaiting trial on sex-trafficking charges in August 2019. He had previously been given an 18-month jail term in 2008 after pleading guilty to soliciting sex from girls as young as 14. After leaving prison, Mandelson continued to have dealings with Epstein, who referred to the politician as “Petie”.

Starmer said the victims of Epstein were “at the forefront of our minds” and described him as a “despicable criminal who committed the most heinous crimes and destroyed the lives of so many women and girls”.

“The ambassador has repeatedly expressed his deep regret for his association with him. He’s right to do so. I have confidence in him and he’s playing an important role in the UK-US relationship,” the prime minister said.

But Badenoch called for all of Mandelson’s correspondence with Epstein to be released and pressed for more details on how much Starmer knew about the relationship.

“The Daily Telegraph reports today that while Lord Mandelson was business secretary, he brokered a deal with Jeffrey Epstein while he was business secretary, and that this occurred after Epstein had been convicted of child sex offences. Given this new information, does the prime minister really think it is tenable for our ambassador to remain in post?” she said.

Starmer said the relationship between the US and the UK “is one of our foremost relationships, and I have confidence in the ambassador in the role that he is doing”.

Davey, the Lib Dem leader, also increased pressure on Starmer by asking whether it was appropriate for Mandelson to remain in his position as the key liaison between the UK and US when the Trump administration may possess further embarrassing material on his dealings with Epstein.

Trump, the US president, is also under scrutiny over his relationship with Epstein, having also allegedly contributed to his 50th birthday book – although he has denied its authenticity and dismissed it as a fake.

Nigel Farage, an ally of Trump and the leader of Reform UK, has recently defended those who had dealings with Epstein in an interview with PoliticsHome, saying: “How could anybody in American high society in New York in the 80s and 90s not have had links with Epstein?

“His business empire his social empire, his tentacle went very, very deep through American society. So the idea that you’re guilty by association because you knew Epstein – I mean you would put the whole of America in jail who were prominent people in business or media in the 1980s.”