In exchanges with Matt Hancock, published by The Telegraph, Mr Case calls then PM Boris Johnson a "distrusted" figure.

He also mocks people forced to stay in quarantine hotels and appears to criticise other minsters.

Mr Sunak rejected calls to sack Mr Case over the comments.

He said the cabinet secretary had "done a great job" and "works hard to support the government's agenda".

Asked if Mr Case would still be in his job at the time of the next general election, Mr Sunak said he looked forward to working with him for a "long time to come".

Separately, a cabinet office insider told the BBC Mr Case was uncomfortable with being in the spotlight, especially as he is unable to respond to the criticism in public because of civil service rules.

Another source said Mr Case's departure from his job was "extremely unlikely".

As a senior civil servant, Mr Case is required to give impartial policy advice to government ministers.

But he has faced criticism over the tone of the exchanges with Mr Hancock.

Conservative MP Marcus Fysh told BBC News the messages "just make it untenable for him [Mr Case] to be credible as a leader of the civil service".

The Yeovil MP, a former member of the Commons Public Administration Committee, said Mr Case should be replaced by someone who can offer a "balanced" view.

"He needs to fall on his sword and the government needs to encourage that," said the MP, who was highly critical of Covid lockdown extensions during the pandemic.

One WhatsApp exchange with Mr Hancock, from June 2020, appears to suggest opposition to tougher Covid rules was "pure Conservative ideology".

In a separate exchange, Mr Case reportedly said then-Chancellor Rishi Sunak had been "going bonkers" over a row about contact tracing.

In another exchange with Mr Hancock, from October 2020, Mr Case said the government was "losing the war" on getting people to isolate after testing because messages were coming from Mr Johnson who was "nationally distrusted".

He added that keeping Mr Johnson focussed on "numbers" of new Covid cases would "keep him honest".

The messages also included exchanges between Mr Hancock and Mr Case, from February 2021, in which the cabinet secretary said it was "hilarious" that international travellers had to isolate on their return to the UK.

"I just want to see some of the faces of people coming out of first class and into a Premier Inn shoe box," he told Mr Hancock.

A former senior civil servant, speaking on condition of anonymity, said the exchanges published by the Telegraph did not necessarily tell the whole story.

But discussing government business on WhatsApp in this way suggested "a lack of professionalism", not least because the messages can be made public.

"It does seem to display a laxity in the decision-making process. with decisions being made for, frankly, emotional, personal, or 'small p' political reasons", he said of Mr Case's exchanges with Mr Hancock.

Some of this may be down to Mr Case's "inexperience" as a senior civil servant, he suggested, and his apparent desire to act as "courtier" to ministers, rather than an impartial adviser.

"As a cabinet secretary, you don't need to curry favour, to be seen to be 'on side' with ministers, in this case Matt Hancock," said the former civil servant.

It was, he suggested, "illustrative, perhaps, of a lack of grip on the job" and he would not be surprised if Mr Case was considering quitting.

"There is wisdom in choosing the time of your own departure and maybe that is something he will be thinking about."

The messages are part of trove of more than 100,000 WhatsApp messages leaked to the Telegraph by journalist Isabel Oakeshott.

The BBC has not been able to independently verify the messages, which were supplied to Ms Oakeshott in confidence by then-Health Secretary Matt Hancock.

Mr Hancock has said they offer an "entirely partial account" of the government's handling of the pandemic.