Sir Keir Starmer has started calling shadow ministers to carry out a reshuffle after hours of wrangling over the role of his deputy, Angela Rayner.
The Labour leader is facing criticism after losing the Hartlepool by-election and control of several councils.
He has already sacked Ms Rayner as the party's chair and campaigns co-ordinator, with sources saying she will be given another role in his team.
But ex-shadow chancellor John McDonnell said she had been made a "scapegoat".
Sir Keir has said he takes "full responsibility" for Labour's election results in England.
Thursday's elections marked the first test of voter opinion since Sir Keir became party leader last year.
With all of England's local authority results in, Labour has lost control of eight councils.
There was bad news for Sir Keir on Saturday night, when a swing to the Conservatives in County Durham saw Labour lose overall control of the council for the first time since 1925.
This followed the loss of the race to become Hartlepool's next MP - held by Labour since the constituency was created in 1974 - to the Conservatives.
There are some really big questions for Labour - but the most straight forward one of all right now is what exactly is going on?
The lights are still on in Sir Keir Starmer's office, but there is still no news about the reshuffle that was meant to happen on Sunday, with no names and no confirmed moves to show how his top team is going to look.
And there is still no clarity in exactly what happened in this dispute with his deputy, Angela Rayner.
You might wonder why that fuss matters at all...
Well, Labour had a hard time this week at those elections, even though they have got a new leader who has made a huge, huge effort to show how much he wants to change the party.
So this moment should have been a really clear chance for him to have a reset, to grab the party by the scruff of its neck and to get on with the changes he wants to make - even if that makes for some very difficult decisions.
But instead, he has found himself in this pretty confused picture - and perhaps a stand off with his deputy - with no clear message about who is even going to be sitting around his shadow cabinet table in the next few days.
It has also provided a vacuum into which his critics - particularly from the left of the party - have spent the whole day touring the broadcast studios, chucking in their tuppence worth, and suggesting he is taking the party in the wrong direction.
The party hopes to give the details of the top team tonight, but it has already been a long old day, with not much clarity about what is going on.
But there was some better news for Sir Keir when the party retained the mayoralties of London, Liverpool and Greater Manchester - and took the West of England mayoralty from the Conservatives.
These results were announced on Saturday, the same day that Sir Keir removed Ms Rayner from her election duties on the same day.
A source said she would "continue to play a senior role" in the leader's team.
But Sir Keir has come in for criticism from senior figures on Labour's left, including Mr McDonnell, who told BBC One's Andrew Marr Show Ms Rayner had been "scapegoated".
He said the leader's style of running the party was "very centralised" and that he had "controlled" the election campaign.
Mr McDonnell also asked what "PR genius" had decided to remove Ms Rayner's campaigning role on Saturday, when there had been some "good news" with the mayoral victories.
He called it a "huge mistake", while former shadow home secretary Diane Abbott described it as "baffling".
And MP Jon Trickett, whom Mr Starmer sacked from the shadow cabinet when he became leader last year, tweeted: "I don't think we should rule out a leadership challenge."
But Dan Norris, who won the previously Conservative-held West of England mayoralty for Labour, said his leader had done "a grand job, a brilliant job" helping with his campaign.
He told BBC Breakfast: "Keir made it possible for people to listen to Labour's message and then be influenced and that's why the result was so good, because people listened to what we had to say... And obviously that's really great for me and great for the Labour Party."
Sir Keir has also been criticised for picking a frontbench team lacking in well-known figures, with suggestions that senior MPs such as Yvette Cooper and Hilary Been could make a return in the reshuffle.
He took over as leader last year following Labour's worst general election defeat since the 1930s and has said the party has a "mountain to climb" if it wants to get back into government.