Sir Keir’s resignation could lead to either a coronation, with Labour MPs electing Mr Burnham unopposed, or an extended leadership contest at which his ideas would be tested.
On Sunday, demands grew for a general election if the leader changes, with Mike Tapp, a Home Office minister, calling for a change in legislation to stop the “constant churn” of politicians. Mr Burnham would be the seventh prime minister in a decade.
Allies of Mr Burnham want a longer transition of power amid concern in his camp that he is unprepared for national government, having spent most of the last decade focusing on local issues.
“He wants to go long, it’s definitely the current idea,” one ally said. “It needs to be at least a month to let each secretary of state meet with the permanent secretaries.
“It would be much like the run-up to a general election, having access talks and seeing how realistic some of the plans are.”
‘Fail to prepare, then prepare to fail’
Clive Lewis, the Norwich South MP and an ally of Mr Burnham, said the former mayor of Greater Manchester should take the time to prepare properly for power.
He told The Telegraph: “Starmer failed to do this. And as my dad always used to drill into me before exams – fail to prepare, then prepare to fail.”
A third Burnham ally said it was “basically in his gift to name a date” but that it would be “reasonable” to delay taking over until the early autumn.
But others expressed concern about the plan, warning that Mr Burnham risked losing momentum and public goodwill during a protracted handover.
“There are mixed opinions but I think the majority want it to happen ASAP,” said one supporter, adding that they wanted to see him in No 10 before mid-July.
It comes amid public disagreements between Mr Burnham’s economic advisers over the critical issue of whether he should increase borrowing, as demanded by some of his Left-wing supporters.
Lord O’Neill, a former Goldman Sachs chief economist and Treasury minister, suggested last week that Britain needed to be “bolder about borrowing to invest”.