Downing Street is undertaking an informal investigation into a “serious” allegation that Sir Gavin Williamson told a senior civil servant to “slit your throat” as the under-fire minister faced a second claim of bullying.

No 10 insisted on Tuesday that Rishi Sunak still maintains confidence in his ally despite the clear unease at the latest accusation as he faces the prospect of three investigations.

An internal inquiry by the Conservative Party is ongoing, while the PA news agency understands that former chief whip Wendy Morton has referred her complaint to Parliament’s bullying watchdog.

Downing Street is understood to be undertaking a fact-finding exercise and considering how to respond to allegations a Ministry of Defence official made to the Guardian newspaper.

The twice-sacked minister denied the official’s allegation of a campaign of bullying while he was defence secretary but did not deny making the extraordinary remarks, including telling the civil servant to “jump out of the window”.

The Prime Minister’s official spokesman said: “There have been further allegations reported this morning. Those are serious allegations that have come in.

“It’s true that no formal complaint has been made but we want to consider proper processes before commenting further.”

It was unclear how the Government would launch a formal investigation, but the Cabinet Office’s Propriety and Ethics team would be one option for handling a complaint.

But Mr Sunak’s spokesman indicated he would not feel the need to wait until the resolution of existing investigations to act.

“I don’t think he necessarily feels that the idea would be for him to wait until both of those things have fully concluded,” the official said.

“Obviously he would act if and when he felt it was appropriate to do so.”

In a statement, Sir Gavin said: “I strongly reject this allegation and have enjoyed good working relationships with the many brilliant officials I have worked with across Government.”

Downing Street did not say whether Mr Sunak has set up any meetings with Sir Gavin, who holds the vague brief of “minister without portfolio” in the Cabinet Office.

The Prime Minister has been coming under sustained pressure over his decision to bring his ally back into Government despite knowing about a complaint made by Ms Morton.

The Times, which first revealed that she has referred her complaint to Parliament’s Independent Complaints and Grievance Scheme, said she made the move over fears of a “whitewash” by the Conservative Party.

In a series of expletive-laden texts, Sir Gavin accused Ms Morton of seeking to “punish” MPs out of favour with then-premier Liz Truss by excluding them from the Queen’s funeral, warning: “There is a price for everything.”

The Liberal Democrats demanded an independent inquiry into Sir Gavin’s conduct so the Tories do not “mark their own homework”.

The party’s Cabinet Office spokeswoman, Christine Jardine, said: “Anything less would be an abdication of leadership from No 10, and make a mockery of Sunak’s promise to govern with integrity.

“The findings must be made public – if the Conservatives have nothing to hide, they have nothing to fear.”

Work and Pensions Secretary Mel Stride said the latest remarks would be “utterly, utterly unacceptable” if proved to be true but said reports are in the “realm of media speculation”.

Any investigation could take several weeks, with Mr Stride suggesting Sir Gavin will still be in post by Christmas as any process “will take some time and we’ve got to wait to see what comes out of that”.

Mr Stride, who spoke of an “aura or mystique around” Sir Gavin, said no minister is “unsackable”.

“The reality with Cronus is he was much touted but he never actually was released to bite anybody,” Mr Stride told Sky News, in reference to the pet tarantula owned by Sir Gavin while he was chief whip.

Labour has continued to maintain the pressure on the Government, with frontbencher Yvette Cooper accusing the Prime Minister of making “grubby political deals that aren’t in the national interest” to appoint Sir Gavin and Suella Braverman as Home Secretary.

Conservative former cabinet minister Baroness Morgan said she had “run-ins” with Sir Gavin when he was Theresa May’s chief whip, adding: “None of this surprises me, sadly.”

“Unfortunately, Gavin has a reputation, it’s not a very nice one, and I really don’t know why Rishi Sunak felt he had to have him back in Government,” she told TalkTV.

Sir Gavin, who was knighted after being nominated for the honour by Boris Johnson earlier this year, is a divisive figure at Westminster, where he is viewed with suspicion by many Tory MPs because of his reputation as an inveterate plotter.

He was sacked first by Mrs May as defence secretary in 2019 for leaking details of a National Security Council meeting, and then by Mr Johnson as education secretary over the Covid-19 A-levels debacle.

However, he was regarded as a key figure in Mr Sunak’s campaign over the summer to become party leader.