"Today, I am putting the people I represent and the country I love first and will be resigning as MP for Makerfield," Simons wrote on X.
"I am standing aside so that Andy Burnham can return to his home, fight to re-enter Parliament, and if elected, drive the change our country is crying out for."
Earlier on Thursday, Labour MP Wes Streeting resigned as health minister and called for a leadership contest to oust Starmer.
A source close to Streeting said he had the numbers to mount a formal leadership challenge but he had not triggered an immediate contest because he felt it would be preferable for Starmer to set out an orderly timetable.
Starmer has repeatedly said he would battle to keep his job, and sources close to the leader say he is determined to fight any contest, which could result in him being challenged by Streeting and also by senior ministers on the left of the party.
Simons cannot transfer his seat to Burnham.
Instead a by-election would have to be held to replace him, in which other parties can also compete.
Burnham said on Thursday he would request permission to stand in a by-election in the region.
"I can confirm that I will be requesting the permission of the NEC (National Executive Committee) to stand in the Makerfield by-election," Burnham said in a post on X.
"There is only so much that can be done from Greater Manchester. Much bigger change is needed at a national level if everyday life is to be made more affordable again.
Simons won the Makerfield seat in Greater Manchester at the 2024 election with a majority of 5399 votes over a candidate from Nigel Farage's Reform UK party.
Since that vote, Reform's popularity has risen sharply as Labour's has dipped, meaning the seat will likely be tightly contested.
A Labour backbencher told the Press Association Burnham would have a good chance despite Reform UK's recent success in the area.
"Game on," the MP, a supporter of Burnham, said.
"I think if we can get everything else to align, we have a fighting chance."
If Starmer does not step down, any challenger for his position would need support from a fifth of Labour MPs, or 81, to trigger a leadership contest.
with PA
