It came as it also emerged that former Home Secretary Amber Rudd has filed a formal complaint in which she accused Sir Philip of being "absent" during the Windrush immigration scandal that led to her resignation.

A Tory source said: "It's quite clear what's going on here. Rutnam is attempting to discredit a Home Secretary who is shaking things up a bit to try and salvage what little is left of his reputation. You can't blame him for trying given that he managed to throw Amber under the bus."

The disagreements became public last week when sources accused Ms Patel of bullying and belittling staff – claims denied by allies of the Home Secretary.

With little sign of the row abating, Ms Patel and Sir Philip attempted to draw a line under the affair on Sunday night by issuing a joint statement in which they criticised "false allegations" and "malicious gossip" and insisted they were focused on delivering the Government’s agenda.

A Home Office spokesman said: "The Home Secretary and Permanent Secretary are deeply concerned about the number of false allegations appearing in the media.

"They are focused on delivering on the Home Office's hugely important agenda, which includes creating an immigration system that works for the UK, putting more police on the streets and keeping the public safe from terrorism."

Earlier, a former insider claimed Sir Philip "spends his time politicking rather than running the actual department."

"You would sit in meetings with Rutnam and Priti would say: 'Where are you with this issue?' and he would say: 'I will have to get back to you on that,'" the former insider said.

Sir Philip was allegedly so obstructive that he was nicknamed Dr No, "because he just said: 'No,'" the source claimed. He was also known for quashing ideas by regularly claiming that there was a 70 per cent chance of the Home Office losing a legal challenge or case.

The insider said the problem was a hangover from Brexit when officials "ran rings round the politicians", adding: "There's a clash now because of them being used to do it their way but suddenly no longer being able to operate that way because of the new regime.

"He is also old school. He got on with Sajid Javid but he had a problem with Amber and Priti. He doesn't get on with strong women who come in with their own views."

The Home Office also described as "categorically untrue" a claim in The Observer that "major run-ins" between Ms Patel and staff had forced the resignation of one of her department's most senior civil servants on immigration.

It said the civil servant had resigned before Ms Patel became Home Secretary.