Priti Patel, the home secretary, has been warned by No 10 to cool down the row with her permanent secretary, Philip Rutnam, after a bitter briefing war between allies of the pair about their deteriorating relationship.

The prime minister’s official spokesman said Boris Johnson had “full confidence” in the home secretary and in the civil service, though the same guarantee was not given to Rutnam specifically.

Behind the scenes, moves have been made to defuse the public spat, with both Patel and Rutnam told to stop the sniping and restore calm in the Home Office.

An ally of Patel said the pair could move on and work together despite their differences, following reports that she tried to oust Rutnam after he raised complaints about her alleged bullying and belittling of staff.

The row further escalated after claims over the weekend that MI5 restricted Patel’s access to security information because they did not trust her – which the intelligence agency and home secretary strongly denied.

It is understood there have been tensions in the Home Office over Patel’s plans for a points-based immigration system, after senior civil servants claimed it was implausible to deliver it within a year.

Friends of Patel claimed that she had tried to request a leak inquiry into the briefing against her from within the Home Office and that it was blocked by the civil service ethics chief, Helen Macnamara. The cabinet office insisted that no such request had been made.

An ally of Patel claimed Rutnam would have to “fall into line” after No 10 gave her its obvious backing. They suggested Rutnam’s position could come under further pressure in the coming weeks, when the long-delayed report into the Windrush deportations – which was due last March – is published. It emerged over the weekend that Amber Rudd, the former home secretary, was unhappy with the level of support from Rutnam when she was under pressure over the scandal.

Over the weekend, the Home Office downplayed the idea of tensions. It released an unusual joint statement from Patel and Rutnam on Sunday expressing concern about “the number of false allegations appearing in the media” and adding that the pair were focused on delivering their department’s “hugely important agenda”.

Intelligence sources on Monday reacted angrily to reports that MI5 was withholding security information from Patel. They said nobody in the spy community could have been responsible for the hostile briefing, implying that any leak must have come from elsewhere in Whitehall.

The initial report claiming that spy chiefs do “not trust” their ministerial boss and “rolled their eyes” in meetings with her was carried in the Sunday Times without any comment from the spy agency, prompting panicked officials to release a rare statement to the newspaper on the morning of publication. A security source belatedly said that Patel was “briefed daily in intelligence matters in exactly the same way as any previous post holder” and that “no information was being withheld” by MI5.

Spy chiefs resent being dragged into political disputes, and sources insist the system of engagement between MI5 and the home secretary is carefully designed to avoid leeway in how it operates in practice. One source familiar with the system said: “I think it would be very hard to suppress information to any home secretary on a systemic basis.”

As home secretary, Patel receives a threat level update and has a meeting with the MI5 director general, Andrew Parker, every week. She also has to sign off surveillance warrants, in addition to a judge. Requests for secret monitoring are typically accompanied by several pages of documentation.

However, Patel has no right to be briefed on the detail or progress of an existing MI5 investigation, and Parker is keen to enforce the semi-autonomy of MI5 within Whitehall. One critic said Parker was “big on the independence of the agency, but then goes out and lets briefings like this happen. He has to be careful he is not having it both ways.”

James Brokenshire, a Home Office minister, said the reports about Patel being denied intelligence briefings were “absolute nonsense”. “Yes, she is demanding, but in that role you have to be because you are dealing with some of the most sensitive, some of the most challenging things that you have to deal with across government,” he told Sky News.

The Conservative former cabinet minister Theresa Villiers said there was an element of misogyny in the “spiteful” briefings against the home secretary – a charge echoed by an ally of Patel. “I think she’s a highly effective home secretary and I think whoever is making these briefings should stop it because I think they are unfair and they are damaging,” they said.