Streeting has said he was releasing his own messages to dispel suggestions he had “something to hide” over his relationship with Lord Mandelson.
Asked on Tuesday if he backed the prime minister, the health secretary said Sir Keir had his “full support” and was leading the country with “integrity”.
Burnham, who along with Streeting has been touted as a possible leadership contender, would not be able to challenge Sir Keir without becoming an MP and his application to stand in the Gorton and Denton by-election was seen as the first step in making a leadership bid.
At the time, Burnham insisted he wanted to return to Parliament “to support the work of the government, not undermine it” but Labour’s National Executive Committee denied him permission to stand saying it would mean “an unnecessary mayoral election”.
Asked on Tuesday if Sir Keir had his full support, Burnham replied: “Yes, he has my support.
“The government has my support and they had my support when I put myself forward for the by-election.”
He called for “stability” in the Labour Party adding: “Of course stability comes from greater unity and that would be helped by a more inclusive way of running the party but recent events makes that now feel possible.”
Elsewhere, the prime minister received backing from the Welsh First Minister Eluned Morgan.
Having stayed quiet on Monday, she put out a statement saying “the country needs stability in an age of instability”.
However, she added that the row over the appointment of Lord Mandelson had been “deeply troubling” and that the “failure must be acknowledged and confronted honestly”.
Reflecting on the events of Monday, Energy Secretary Ed Miliband told BBC Breakfast, the Labour Party had “looked over the precipice” on Monday but “thought the right thing to do was to support our leader”.
Miliband – a former Labour leader – said Sir Keir had faced “a moment of peril” but that cabinet ministers had collectively “looked at the alternatives of going down this road of a chaotic leadership election, trying to depose a prime minister, and they said that’s not for us”.
