A top ally to Prime Minister Keir Starmer of Britain refused on Sunday to repeat Mr. Starmer’s promise to fight any challenge to his leadership of the Labour Party, a notable shift in tone over whether the prime minister would step aside.

That ally, Peter Kyle, the government’s business secretary and a close friend of the prime minister’s, told the BBC that Mr. Starmer was “taking the time to think through what the political realities are today compared to last week.”

He added that Mr. Starmer was “making time this weekend to try and reflect on the political challenges that he faces” and that the prime minister and the party “are going to put the country first.”

Mr. Starmer’s challenges include an imminent bid by Andy Burnham, the former mayor of Greater Manchester, to oust Mr. Starmer from his position as prime minister. Mr. Burnham won a Labour Party seat in Parliament last week after a special election on Thursday in Makerfield, in northwest England, and has made it clear that he wants Mr. Starmer to step aside so he can take over.

Mr. Kyle’s comments on Sunday are significantly different from Mr. Starmer’s own remarks 48 hours earlier, when said he would fight to keep his job, no matter what.

“If there is a contest,” Mr. Starmer told reporters on Friday morning, “I will stand, and I’ve said repeatedly I’m not going to walk away.”

More than 100 Labour Party lawmakers have already called on Mr. Starmer to set out a reasonable timeline for his resignation. Reports over the weekend suggested that more could do so when Parliament returns on Monday.

Speaking to supporters on Friday on a Labour Party conference call, Mr. Starmer urged his parliamentary colleagues not to embrace a leadership contest that could undermine the standing of the party and damage efforts to effectively govern Britain.

“The one thing we’ve got to avoid doing is plunging our party and our country into chaos by turning on each other and tearing apart our party,” he said. “That has never worked.”

Two senior aides to Mr. Starmer said late Saturday, and again on Sunday, that the prime minister stands by his statements from Friday morning about fighting any challenge. They said he was spending the weekend doing the business of governing the country.

But Mr. Starmer’s standing as the head of Britain’s government has been declining for months amid economic struggles, political scandals and unpopular policy choices. Last month, the Labour Party lost nearly 1,500 seats on local municipal councils, with many voters saying they blamed Mr. Starmer for the struggles they face.

Mr. Burnham’s successful move to secure a spot in Parliament was devised specifically to give him the opportunity to challenge Mr. Starmer. In a speech on Friday after his victory, Mr. Burnham pledged to move quickly to capitalize on his political momentum.

“It is our last chance to change, but we’re going to take it, aren’t we?” he said. “We are going to take that opportunity, and we are going to lay out a new path for Britain.”

Appearing on Sunday on the BBC’s “Sunday With Laura Kuenssberg,” Mr. Kyle repeatedly said that Mr. Starmer would ensure that any decision he made would be “in the best interests of the country.” And he said that the prime minister wanted to ensure that the authority of Britain’s government was not affected by political strife.

“I don’t want to come on here and be delusional that there is no process, there is no forces at work which are challenging the Prime Minister as leader. That is clearly the case,” Mr. Kyle said. “What I’m trying to say is that it is really important that we govern through whatever unfolds or may not unfold in the days ahead.”

Many Labour Party members have called on Mr. Starmer to outline an “orderly transition” by announcing a timeline for his resignation that might avoid a nasty leadership contest. Mr. Kyle refused to use those words, but his comments appeared to acknowledge the need to avoid a damaging political fight.