There is speculation Welsh Language Minister Eluned Morgan could announce she wants to be the next Welsh Labour leader in a statement later.
But senior Labour figures have told BBC Wales they doubt Ms Morgan has the support in the party’s assembly group to be sure she can run in the contest.
Finance Secretary Mark Drakeford is the only declared candidate so far.
Ms Morgan’s office said she will speak about the leadership at a money advice event in Pembroke Dock on Friday.
A number of Labour politicians have said they want a diverse range of contenders, including a female candidate, and warned against a “coronation”.
BBC Wales also understands the Economy and Transport Secretary Ken Skates, AM for Clwyd South, is under intense pressure to run to ensure a politician from north Wales stands.
Candidates need support from five other Labour AMs to be on the ballot paper. Mr Drakeford is backed by eight AMs – more than enough to run.
Speaking on Thursday, Ms Morgan, a life peer and Mid and West Wales AM, said any leadership contender would need an answer to problems such as “in-work poverty” and “other fundamental questions that impact on people’s daily lives”.
She also said she was taking “time to reflect” on a possible bid.
Carwyn Jones announced at last weekend’s Welsh Labour conference that he will step down as first minister and the party’s leader in Wales after the autumn contest.
Analysis by Nick Servini, BBC Wales political editor
Ordinarily you would assume this is the moment when someone is going to declare a candidacy.
But at this point it does not look like Eluned Morgan has the numbers.
The statement could easily be about what kind of contest she would like, or even announcing her intention to stand without necessarily having sufficient nominations.
There is also a strong view among some AMs that despite the fact Mark Drakeford came out of the traps early, there is no reason to rush through a process in which the party ends up with a slate of candidates.
After all, when Rhodri Morgan announced his departure timetable there was plenty of time before a final list of candidates emerged.
We are heading into a weekend where a lot of Labour AMs have plenty to think about.