Sir Keir said he had asked Labour’s governing body to set out a timetable to replace him, with nominations opening on 9 July and ending by the summer recess on 16 July.
Potential candidates have until then to amass the support of at least 81 Labour MPs.
Burnham is seen by many as the frontrunner to replace Sir Keir after an emphatic victory over his Reform UK rival in last week’s Makerfield by-election.
The former Greater Manchester mayor, who was formally sworn in as an MP in the House of Commons on Monday, said he would put himself forward as a candidate in the leadership contest.
Wes Streeting had been viewed as his main rival but the former health secretary was quick to offer his support to Burnham, increasing the likelihood of a “coronation”.
John Slinger, the Labour MP for Rugby, voiced the feelings of many within the party who believe it is “necessary” to have a full leadership contest.
“I think the public out there would think we’d slightly lost our minds if we didn’t go through a process where we subject people who aspire to the highest office in the land to completely normal scrutiny,” he told the BBC’s The World Tonight on Monday.
Describing Sir Keir as a “terrible prime minister”, Conservative Party leader Kemi Badenoch attacked the Labour Party’s policies, saying: “Labour MPs only want higher taxes to hand out more benefits… These are Labour’s choices and their values, regardless of who is running the party.”
Liberal Democrat leader Sir Ed Davey said the British people were “sick of being let down by an endless merry-go-round of prime ministers while nothing really changes”.
Reform leader Nigel Farage demanded a general election, saying: “If Labour thinks it can shove another professional politician into No 10, it has another thing coming.”
Zack Polanski, leader of the Green Party of England and Wales, said people would hope that Burnham could bring about “meaningful change” but the “jury is out”.