Solemn, controlled and with the delivery of the top-rank lawyer he once was, Prime Minister Keir Starmer of Britain was out early on Friday, pointing out that no formal leadership challenge has yet been triggered against him and pledging to fight to keep his job.

Some 200 miles north, his Labour Party rival Andy Burnham promised change and urgency as he addressed jubilant supporters who helped him win a special election. Declaring a new era of hope, Mr. Burnham, whose victory means he is now eligible to challenge the prime minister, said that Labour faced its “last chance to change,” adding: “We are going to take it.”

The stark contrast between the two politicians’ styles helps explain why many within Labour now believe Mr. Burnham should replace Mr. Starmer, who has become one of the most unpopular prime ministers in British polling history.

Pressure is now rising on Mr. Starmer to agree to a timetable to relinquish power and avoid a divisive battle for his job.

Mr. Burnham, who has twice tried but failed to become Labour leader, has made no secret of his ambitions to lead his party and the country. With his stunning special election victory in Makerfield, in northwest England — where he won a majority of about 55 percent — he appears to have demonstrated that he has the charisma and star quality to take on Reform U.K., the populist right-wing party that has upended British politics.

For the past two years, Reform’s rising popularity in the polls under the leadership of Nigel Farage, an ally of President Trump and an architect of Brexit, has terrified its opponents.

The mechanics of what happens next, however, are hazy.

“Burnham doesn’t want to be seen to be politically assassinating Starmer,” said Steven Fielding, emeritus professor of political history at the University of Nottingham. “He wants him to go — at least ostensibly — under his own steam.”

Mr. Starmer would lose if he decided to run in a leadership contest against Mr. Burnham, Prof. Fielding argued, and the process would divide and damage the Labour Party. “It would be like Custer’s last stand: Nobody wins, the party won’t win, the country won’t win and he won’t win.”

To trigger a leadership contest, a challenger needs to be a Labour lawmaker — a hurdle Mr. Burnham has now cleared — and have the support of 20 percent of their colleagues, or about 80 lawmakers currently. Under the Labour rulebook, fee-paying party members would get the final choice. However, if Mr. Starmer agreed not to fight and Mr. Burnham was the only candidate, the membership would not have to vote.

Wes Streeting, who recently quit the cabinet as health secretary, has indicated he wants to run for leader, though opinion polls show him to be significantly less popular than Mr. Burnham.

An alternative to a leadership race could be for Mr. Starmer to step aside, perhaps in time for Labour’s annual conference in September, and for Mr. Streeting to accept a senior cabinet position under Mr. Burnham in exchange for agreeing not to force a contest.

On Friday morning, Louise Haigh, a Labour lawmaker and close ally of Mr. Burnham, said she hoped the process would be “as managed and controlled as possible.”

“I would hope that Andy and the prime minister can speak over the coming days,” Ms. Haigh told the BBC. “We want to avoid a leadership contest if possible, and we hope that we can agree a way forward.”

So far, Mr. Starmer has given no indication that he will bow to pressure and step aside. However, while leading the party in opposition in 2021, he once briefly contemplated resigning for the good of the party after Labour lost a special election in Hartlepool.

He later told Sky News he considered quitting “because I didn’t feel that I should be bigger than the party and that if I couldn’t bring about the change, perhaps there should be a change.”

One way of putting pressure on Mr. Starmer would be for ministers to threaten to resign from his government, if he refuses to go. In 2022, mass ministerial resignations forced the Conservative prime minister, Boris Johnson, from Downing Street, by making his government untenable.

If he is confident enough of his support, Mr. Burnham could effectively tell Mr. Starmer to choose how he wants to wrap up his time as prime minister: the easy way or the hard way.

In either case, the scale of Mr. Burnham’s victory has left him in a commanding position.

In the fight for Makerfield, Reform U.K’s candidate, Robert Kenyon, faced competition from a new far-right party called Restore Britain. Some analysts expected that to split the vote, letting Mr. Burnham through by default. But Mr. Burnham’s vote total easily exceeded that of both Reform and Restore combined.

Some factors helped Mr. Burnham who, as mayor of Greater Manchester, had established himself as a popular local politician. Because of his well known ambitions to take over the Labour Party, some voters may have opted for him in the hope of getting rid of Mr. Starmer.

On Friday, Mr. Farage said he was “disappointed” with the outcome, but said Labour had benefited from running a campaign that urged voters: “Vote Burnham, get Starmer out.” Reform, Mr. Farage argued, was successful in many municipal elections in May after arguing that voting for his party would help get rid of Starmer, so “we were slightly hoist with our own petard.”

But the loss in another special election will prompt soul-searching in Reform, which saw Makerfield as a stronghold.

Reform has led other parties in national opinion polls for more than a year, with about 30 percent saying they supported it last year. That share has dipped recently to around 25 percent.

Reform lost a special election in February to the Green Party in the Gorton and Denton district in northern England, and it lost another one last year in Caerphilly to Plaid Cymru, a center left Welsh nationalist party, in a contest for a seat in the Welsh Parliament.

“If we can’t win in Makerfield, can we win anywhere?” Tim Montgomerie, a former Conservative who defected to Reform, pondered in an interview with The New York Times last month.

The result may indicate, some analysts say, that Reform’s agenda is so polarizing that a majority of British voters will opt for whatever party they believe can defeat it in any specific vote, even if it’s not their first choice.

For now, Mr. Burnham, who took a gamble by running in the Makerfield seat, has hit the political jackpot.

“The way that Burnham won has created an even greater sense of momentum and inevitability, that’s he’s the one that can save the Labour Party,” said Mr. Fielding. “It’s in the Labour Party’s interests for this to be resolved without a leadership contest by the end of next week.”