The Greater Manchester mayor is standing in the Makerfield by-election, in a high-stakes attempt to re-enter the House of Commons.
Asked if Sir Keir should go, he said: “Yes, I think he should, but I think he should do it in a planned way, in a dignified way in a way that allows for a smooth transition.
“I think it is irrecoverable both his own personal standing and his ability to lead an effective government on the doorstep.
“I think, quite unfairly, there was a level of hostility towards the prime minister, which I don’t think can be recovered.”
He added: “We need somebody who speaks authentically for the wider United Kingdom, the regions of England, the nations of the United Kingdom.
“I think Andy is uniquely best to do that, and that’s why he will have my strong support.”
If Burnham did not win, Drakeford said there would “be another candidate that represents the sorts of views that Andy represents”.
Appearing to support Angela Rayner in that scenario, he added: “If Angie were to be the standard bearer of that strand, in the party’s opinion, I would be very keen to support her too.”