The Greater Manchester mayor will first have to overcome a strong challenge from the anti-immigrant Reform UK party in a special election for the parliamentary seat that was vacated to make way for him.
British government borrowing costs rose on Friday and the pound weakened on investor concern about continued disarray at the heart of government.
Weeks of speculation about Starmer 's future broke into open rebellion within the governing Labour Party on Thursday as Burnham declared his intention to seek the top job and two other senior members positioned themselves for their own bids.
The pressure to replace Starmer increased after Labour posted disastrous results in local elections on May 7, losing votes to Reform UK on the right and the Green Party on the left.
Housing Secretary Steve Reed on Friday appealed to party members to step back from the brink of a divisive leadership contest that he said would prevent the government tackling issues such as a cost-of-living crisis and bolster the prospects of Reform UK.
"This weekend people just need to take a breath, look at what's gone wrong this week, and come back next week ready to do what we said we'd do - country first, party second - and focus on delivering the change we were elected to deliver," he told the BBC.
That plea came after a week in which political jockeying overshadowed everything else in Westminster.
After dozens of Labour members publicly called for Starmer to step down, Health Secretary Wes Streeting on Thursday became the first cabinet minister to resign.
While praising Starmer's "courage and statesmanship" in international affairs, Streeting said he had lost confidence in the prime minister's leadership because of missteps on domestic issues.
"Where we need vision, we have a vacuum. Where we need direction, we have drift," Streeting wrote in a stinging resignation letter.
"Leaders take responsibility, but too often that has meant other people falling on their swords.
"You also need to listen to your colleagues, including backbenchers, and the heavy-handed approach to dissenting voices diminishes our politics."
Streeting stopped short of putting himself forward as the best candidate to lead the party at the next general election, suggesting Starmer should step aside to allow a "broad" field of candidates to debate the future of the party.
That seemed to be a nod to Burnham, a former cabinet minister who left parliament in 2017 to run for mayor of Greater Manchester.
Burnham has been looking for a way to return to the House of Commons so he can challenge Starmer for the top job.
Josh Simons, a Labour MP from northern England, provided that opening on Thursday by resigning his seat explicitly to open up a seat for Burnham.
But that was only the first step for Burnham.
Before he can return to Westminster, Burnham must win a special election to represent Makerfield, a community where Reform UK posted strong results in local elections.
Burnham acknowledged these challenges on Thursday when he announced his candidacy for the seat.
"I truly do not take a single vote for granted and will work hard to regain the trust of people in the Makerfield constituency, many of whom have long supported our party but lost faith in recent times," he said in a statement.
