Burnham confirmed his intention to stand shortly after Sir Keir stepped down as prime minister on Monday.
There had been suggestions that senior minister Darren Jones or former Armed Forces minister Al Carns could try to stand – however Jones has ruled himself out and Carns is yet to confirm if he will make a bid.
Former Health Secretary Wes Streeting had previously said he would throw his hat into the ring, but gave his backing to Burnham on Monday.
Burnham has also received backing from cabinet ministers including Chancellor, Rachel Reeves, Education Secretary Bridget Phillipson and Health Secretary James Murray. He also looks set to easily clear the 81 MP threshold.
It is looking increasingly likely that he will gather so many nominations no other Labour MP will be able to stand.
Sir Keir has said he wants an “orderly” transition of power and has authorised civil servants to meet prospective Labour leadership candidates to help them prepare for government.
Despite announcing his resignation, Sir Keir will stay as prime minister until a new Labour leader is elected.
Earlier, his official spokesperson denied that the administration is now a “zombie government”.
“The prime minister and all ministers remain in office and the normal business of government continues,” the spokesman said.
“That includes domestic priorities, such as public services, economic stability and reforms, alongside international leadership on security and growth, and you saw evidence of that yesterday with the PM in Berlin, and the focus very much remains on serving the country and getting on with the job”.
Downing Street has also said it will not make “major polices” or spending decisions during the transition to the next prime minister.
However, it is still planning to publish the long-awaited Defence Investment Plan (DIP) which sets out spending for military equipment, before the Nato conference on 7 July.
Sir Keir plans to attend the defence alliance’s conference in Turkey in what is likely to be one of his final official duties as UK prime minister.
