Senior Labour figures have criticised calls for a new parliamentary investigation into whether the prime minister was honest with MPs about the vetting process for Lord Mandelson's appointment as the UK's ambassador to the US.

Conservative leader Kemi Badenoch said Sir Keir Starmer had misled Parliament "multiple times" on the subject.

She urged Labour MPs to "look into their consciences" and back a new inquiry by the Privileges Committee.

Labour MP Dame Emily Thornberry, who chairs the Foreign Affairs Committee which is looking into the appointment, told the BBC there was no "rush" to set up a new inquiry and suggested some MPs were "trying to score points ahead of the local elections".

Separately ex-Labour ministers Lord Blunkett and Alan Johnson have said an inquiry would be a "waste of money".

The Commons Speaker Sir Lindsay Hoyle would need to allow a debate on the matter and it would be up to MPs to vote on whether to launch an investigation.

If he does allow a vote, this could take place as soon as Tuesday.

The government holds a majority in the House of Commons, so a large number of Labour MPs would have to vote for an inquiry or abstain in order for one to be launched.

The Privileges Committee can look into cases of MPs breaking parliamentary rules and in 2023 it ruled that the former Prime Minister Boris Johnson had misled MPs about parties in Downing Street during Covid.

The Ministerial Code states that ministers who knowingly mislead Parliament are expected to resign, while any inadvertent error should be corrected "at the earliest opportunity".

Lord Mandelson was sacked seven months after starting the Washington DC job over his friendship with the convicted sex offender Jeffrey Epstein.

Sir Keir apologised for making the appointment but has continued to face questions over whether the vetting process was rushed.

Speaking on Monday, Conservative leader Kemi Badenoch said there was "still a lot of information that doesn't add up".

"What I'm seeing is a prime minister who is saying whatever he needs to to save his own skin," she added.

She said Sir Keir had misled Parliament when he told MPs that "full due process" had been followed in the appointment process.

She also questioned the prime minister's assertion that "no pressure existed whatsoever" on the Civil Service to approve Lord Mandelson as the UK's ambassador.

However, Sir Olly Robbins, who was the senior civil servant in the Foreign Office until he was sacked by the PM, told the Foreign Affairs Committee last week that there was "constant pressure".

He added that this did not affect his decision to give Lord Mandelson security clearance to take up the role.

In an apparent effort to clarify his comments in Parliament, Sir Keir told the Sunday Times there are "different types of pressure".

"There's pressure – 'Can we get this done quickly?' – which is not an unusual pressure. That is the everyday pressure of government," he said.

Defending the prime minister, Environment Secretary Emma Reynolds said: "It was categorically proven last week that the PM did not lie to Parliament.

"The prime minister had said that due process was followed and due process was followed, unfortunately the process was fundamentally flawed."

A vote on launching a Privileges Committee inquiry is likely to take place on Tuesday - the same day that senior former government figures are due to give evidence to the Foreign Affairs Committee including the prime minister's former chief of staff Morgan McSweeney and former senior civil servant at the Foreign Office Sir Philip Barton.

Liberal Democrat leader Sir Ed Davey said: "Labour MPs must be given a free vote on any motion to refer Starmer to the Privileges Committee, not forced into being accomplices to a cover-up.

"If Keir Starmer has misled the House and the public, he must be held to the same standard that we should expect of any prime minister."

Reform UK's economic spokesman Robert Jenrick said the prime minister had "misled Parliament" and had "behaved appallingly".

However, he added: "Parliament needs to actually talk about the priorities of the British people, and that is not primarily about the ins and outs of Peter Mandelson."

In a joint statement to the Times, former Labour ministers Johnson and Lord Blunkett said the Conservative calls for a Privileges Committee inquiry were a "nakedly political stunt".

"Any comparison with Boris Johnson is absurd. When Parliament referred that matter to the Privileges Committee, a police investigation had directly disproved his categoric statements that he knew nothing about the breach of lockdown rules," they said.

Speaking to the BBC Radio 4's Today programme, Dame Emily Thornberry said her committee was investigating the appointment and that she did not want the Privileges Committee to be "duplicating the work we're doing".

"It may be that at some stage in the future, some of the questions haven't been answered, and it is decided that they are of sufficient importance that the Privileges Committee should be involved," she said.

"But I don't really see why we're doing it at the moment, apart from, potentially people trying to score points in advance of the local elections.

"I'm sorry to say that, and I'm not supposed to be partisan on this, but it is as plain as the nose on my face what's going on here."

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