"Someone has definitely been watching too much Celebrity Traitors. They should swap to Countryfile," he said.
Streeting was doing a round of media interviews on a morning, with headlines dominated by the briefings from the prime minister's allies.
The health secretary insisted he was "not challenging the prime minister" adding: "I'm not doing any of the things some silly briefer said overnight."
He said the culture in No 10 was "toxic" and that Lucy Powell, the party's new deputy leader, had been right to call for a change of culture in Downing Street.
"I'd like to commend the briefer on at least picking on one of the men in the cabinet instead of the women," he added.
Asked if he should sack those responsible, Streeting said: "Yes. But he's got to find them first and I wouldn't expect him to waste loads of time on this.
"There are people around the prime minister who do not follow his model and style of leadership."
