Sir Keir Starmer has insisted he has “never authorised” attacks on his cabinet ministers, calling briefings against them “unacceptable”.

The PM was speaking at Prime Minister’s Questions the day after his allies told numerous media outlets he could face a leadership challenge from one of his colleagues, including Health Secretary Wes Streeting.

Conservative leader Kemi Badenoch accused the prime minister of running a “toxic” Downing Street and said it proved Sir Keir had “lost control of his government… and lost the trust of the British people”.

Earlier, Streeting dismissed the claims of a challenge as “self-defeating nonsense” and said those behind the anonymous briefings should be sacked.

The prime minister said his focus “each and every day is on rebuilding and renewing our country”.

“Any attack on any member of my Cabinet is completely unacceptable.”

He added that Streeting – who was not in the House of Commons but in Manchester giving a speech on the NHS – had got waiting lists down.

“He’s doing a very good job,” he added.

Badenoch said Morgan McSweeney, the Downing Street chief of staff, was responsible for the culture in No 10 and asked if the prime minister still had confidence in him.

Sir Keir replied: “Morgan McSweeney, my team and I are absolutely focused on delivering for the country.

“Let me be clear, of course, I’ve never authorised attacks on cabinet members, I appointed them to their post because they’re the best people to carry out their jobs.”

Following the Commons question session, the prime minister’s press secretary told reporters the briefings against Streeting came “from outside No 10” and that the prime minister had full confidence in the chief of staff Morgan McSweeney.

The spokesperson refused to say whether there was a leak inquiry, but did say leaks would be “dealt with”.

Briefings of this nature are often part of reporting on politics in Westminster, when people speak to journalists “off the record”.

There is often a big difference between what someone will say on camera and what they are saying behind the scenes, and as a result, stories can refer to “friends of” or “sources close to” or “briefings” or sometimes “the BBC understands”.

It can make Westminster sound like a secret society, but if the BBC just reported things that were on the record, we would only be telling half the story.

On Tuesday evening, supporters of the prime minister told journalists he would fight a challenge to his leadership, which they believed could come as soon as after the Budget on 26 November.

They argued that removing the PM could create chaos, destabilise the international markets and damage the relationship he has built with US President Donald Trump.

The names being discussed by Labour MPs as potential candidates to replace Sir Keir include Home Secretary Shabana Mahmood. There is also speculation Energy Secretary Ed Miliband could stand.

Those close to the prime minister are believed to be particularly suspicious of Streeting.

The health secretary had a chance to respond to the claims when he was doing media interviews on Wednesday morning.

Speaking to the BBC, he said he was “not challenging the prime minister” adding: “I’m not doing any of the things some silly briefer said overnight.”

Asked if he could consider standing against the PM in the future, he replied: “I cannot see circumstances in which I would do that to our prime minister.”

He said the briefings to the media demonstrated that the culture in Downing Street was “toxic”.

He suggested he did not believe the prime minister had approved the briefing, telling the BBC: “I can tell you without having even spoken to the PM what he thinks of the briefings – the words he would use are not suitable for a family show like this.”

Other Labour MPs and some ministers have publicly and privately condemned the briefings.

Appearing on BBC Politics Live, Business Minister Sir Chris Bryant said he thought they were “plain daft”.

Labour MP for Bassetlaw Jo White said: “This is a group of people who think they’re much cleverer than the rest of us, who spend their time selectively briefing journalists and stirring the pot.

“I want to simply say: we’re not having it.”

Some Labour MPs who are usually supportive of the prime minister described the timing and substance of the briefing as “unhinged”, “badly handled” and “baffling”.