It comes as Sir Keir decided to withdraw Lord Mandleson as Britain’s man in Washington on Thursday after emails were published allegedly showing Lord Mandelson sent supportive messages to Jeffrey Epstein as the disgraced American financier faced jail for sex offences, piling further pressure on his embattled government.

Emails published on Wednesday afternoon included passages in which Lord Mandelson allegedly told Epstein to “fight for early release” shortly before he was sentenced to 18 months in prison, the newspaper reported.

He is also alleged to have told Epstein “I think the world of you” the day before the disgraced financier began his sentence for soliciting prostitution from a minor in June 2008.

The PM is feeling the heat after the second scandal-hit departure for the Government in a week after Angela Rayner quit over her tax affairs. Labour MPs are beginning to ask whether Sir Keir should face a leadership challenge, and have warned that time is running out to turn things around before the local elections in May, The Guardian reports.

One MP said a potential bid from Burnham is the “talk of the tearoom” in Parliament, and that Mainstream, which is made up of members critical of Sir Keir, was interpreted as a challenge to the PM, The Telegraph reports.

Speaking to the newspaper, they said: “What has accelerated its development is that the Labour Government is in trouble with the public, and there is now a king over the water, in the shape of Andy Burnham. This is really about Andy setting his stall out to become leader of the Labour Party.”

According to The Telegraph, Burnham is expected to explicitly criticise Sir Keir at the upcoming Labour party conference and call for a “reset” to help Labour retain power at the next general election. His support for Lucy Powell in Labour’s deputy leadership race has also been interpreted as an attempt to have an ally in the party’s senior leadership, the outlet reports.

A spokesperson for Mainstream reportedly didn’t deny that it could become a leadership vessel for Burnham.

However, Burnham has previously pledged to see out his term as mayor, with his tenure set to end in 2028. Rules prevent people from serving as mayors and being MPs (who can run for PM) at the same time.

Insiders told the paper that Burnham could look to compete in a by-election ahead of the next election.

The Prime Minister is understood not to have been aware of the contents of the alleged emails between Lord Mandelson and Epstein until Wednesday evening – after he told the Commons he had “confidence” in the Labour grandee during Prime Minister’s Questions at midday.

Downing Street and the Foreign Office said the emails showed “the depth and extent” of Lord Mandelson’s relationship with Epstein was “materially different from that known at the time of his appointment”.

The emails were sent from an account which had long been closed and were not available during the vetting process.

Allies of the peer claimed to The Times that he admitted in his vetting interview that he continued his relationship with Epstein for many years.

Sir Keir and the Foreign Office are facing questions over what they knew and when about the ex-ambassador’s ties to Epstein.

The Cabinet Office has been approached for comment.