They added: “I can categorically say he was not involved indirectly or directly.”
Journalists were briefed on Tuesday by allies of the PM that he would take on any challenge to his leadership, in an attempt to shore up his position.
There were also briefings that he faced a threat from named ministers, particularly Streeting.
Government sources have sought to draw a distinction between the two.
Sir Keir earlier said he had been assured by his team in Downing Street that the briefings about cabinet ministers did not come from Number 10.
Speaking during a visit to North Wales, the prime minister said: “I have been talking to my team today. I have been assured that no briefing against ministers was done from No 10 but I have made it clear that I find it absolutely unacceptable.”
He added: “I have been assured it didn’t come from Downing Street but I have been equally clear that whether it is this case or any other, I intend to deal with it.”
Asked if he would sack those responsible, Sir Keir replied: “I will absolutely deal with anybody responsible for briefing against ministers, cabinet ministers or any other ministers.
“I have always said that is the standard that I expect, and that is the standard that I will enforce.”
Asked whether he still has confidence in McSweeney, the Labour leader said: “Of course I do. I work closely with my whole team. Our focus is on working for the country.”
One Labour MP and McSweeney ally told the BBC they were “really worried” about the chief of staff’s welfare amid the torrent of public criticism.
Briefings are often part of reporting on politics in Westminster, when people speak to journalists “off the record” on condition that they are not named.
This means they say things that they are not prepared to say on camera, which the BBC reports in order to give the full story.
