Sir Keir Starmer is weighing up his political future as pressure grows on him to announce his resignation.
It comes as cabinet ally Peter Kyle said the prime minister was considering "political realities" after Andy Burnham's victory in the Makerfield by-election last week cleared a path for him to challenge for the Labour leadership.
In the wake of the result, the prime minister has faced fresh pleas from senior ministers to effectively make way for Burnham by setting out a timetable for his departure.
And in a further blow to the embattled Labour leader, US President Donald Trump said Sir Keir "will resign as prime minister", as he attacked his record in office.
In a social media post, Trump said Sir Keir had "failed badly" on immigration and energy policy, as he repeated his calls to "open North Sea oil".
He added: "I wish him well!"
No 10 told the BBC the two leaders, who last saw each other at a G7 summit in France last week, had not spoken over the weekend.
In the immediate aftermath of Burnham's victory, Sir Keir continued to insist he would fight any formal leadership bid, meaning a contest in which Labour members and affiliated trade union supporters decide on the party's future would be needed.
But in a shift in tone, Kyle, the business secretary, told the BBC's Sunday with Laura Kuenssberg that the prime minister was "taking the time to think through what the political realities are today, compared to last week and the week before".
He said he had spoken to Sir Keir on Friday and was sure that "every decision he makes today" about Labour's future would reflect "what's in the best interests of the country".
Kyle said he did not want to "be delusional that there is no process" and "no forces at work" that were challenging Sir Keir as party leader.
While using a formal process to change leader was "better wherever possible", this needed to be balanced against the need to maintain the government's authority "through any processes that may unfold", he added.
The 2020 leadership race that led to Sir Keir being made Labour leader took six weeks, with some Labour MPs expressing concerns that disagreements that would accompany a similar contest could further damage the party's prospects.
Concerns have also been raised that this could create unnecessary uncertainty for the markets and delay key government decisions.
Burnham was able to fend off a challenge from Reform UK in Makerfield on Thursday and increase Labour's majority, bucking a recent trend of electoral losses.
Transport Secretary Heidi Alexander and Foreign Secretary Yvette Cooper have since urged Sir Keir to set out an exit timetable, it is understood.
Home Secretary Shabana Mahmood and Energy Secretary Ed Miliband had already done the same in the aftermath of devastating results for Labour in a series of elections last month. Scores of Labour MPs have also publicly urged Sir Keir to resign immediately or draw up an exit timetable.
Burnham, who is understood to be spending time with his family this weekend, is expected to travel to Westminster on Monday to formally take up his seat as an MP.
His allies urged the prime minister to reflect over the weekend and listen to his cabinet ministers, MPs and his family.
Former Health Secretary Wes Streeting, a potential leadership rival to Burnham, has previously said he would join any contest, arguing the party needed to undergo a "battle of ideas" over its future direction.
Jess Phillips, who was one of four ministers to quit in the immediate aftermath of May's election results, said: "It feels like we've come to the end of the road."
But the former safeguarding minister, seen as a close ally of Streeting, said she hoped the party would find a means to question "what's coming next" even if this did not "end up with a full-scale contest".
"You can't just come and take over," she told Kuenssberg. "You do have to come and present your ideas to, at the very least, the Parliamentary Labour Party."
During the month-long Makerfield campaign, Burnham recommitted to Labour's manifesto promise not to increase the main rates of income tax, VAT or National Insurance, as well as the borrowing rules set by Chancellor Rachel Reeves.
He has expressed a desire for "stronger public control" of utilities and repeated previous pledges to replace inheritance tax with a new "national care levy".
However, he has yet to set out his thinking in other policy areas, including on defence spending, while Sir Keir has of late focused his efforts on rewriting government budgets to free up more cash for military investment.
Labour MP Toby Perkins, a shadow minister in opposition, told Kuenssberg he did not want Sir Keir to stand down, adding it would mean the country would have its "seventh prime minister in 10 years".
He said the prime minister "deserves a bit of time" and that he did not believe the government was "manifestly failing", noting it had lowered NHS waiting times, net migration and the asylum backlog.
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