Angela Eagle looks set to launch a bid for the Labour leadership but a source close to her told the BBC Jeremy Corbyn "still has time to do the right thing".
Mr Corbyn has rejected an attempt by Labour's deputy Tom Watson to persuade him to stand down.
But Ms Eagle, who joined a mass walkout of frontbenchers, is thought unlikely to launch a leadership bid on Thursday.
Mr Corbyn has said he would not "betray" the party members who elected him last year.
Ex-shadow minister Owen Smith is also said to be considering a leadership bid.
Asked if she was launching a leadership bid as she left home earlier, Ms Eagle said: "I will be saying something later today."
But a source close to her has told BBC political editor Laura Kuenssberg: "Jeremy Corbyn still has time to do the right thing."
The source said Ms Eagle will challenge him if necessary but hopes that he will go of his own accord.
Tom Watson and other senior figures fear the party is facing a threat to its existence following an unprecedented vote of no confidence in Mr Corbyn by Labour MPs.
What about Europe?'
Mr Watson told BBC News: "My party is in peril, we are facing an existential crisis and I just don't want us to be in this position because I think there are millions of people in this country who need a left-leaning government."
Labour-supporting trade unions indicated in a statement that they would not stand in the way of a contest if it was carried out "through the proper democratic procedures provided for in the party's constitution".
In a speech to students in London on Wednesday evening, Mr Corbyn said that while he recognised not everybody supported the direction in which he was taking the party, he had the mandate to carry on.
"I also recognise that the mandate was given by hundreds of thousands of ordinary people joining in the political process," he said to cheers from supporters.
He was briefly heckled, with one man shouting: "What about Europe? Where were you when we needed you?"
Ms Eagle, who was drafted into Mr Corbyn's top team as the shadow business minister and first secretary of state last year, is one of 20 members of the shadow cabinet to have quit since Sunday in protest at his leadership.
It is understood she has the support of the 51 Labour MPs and MEPs needed to mount a challenge.
The BBC understands former shadow work and pensions secretary Owen Smith and his "friends" are canvassing support among Labour MPs for a possible leadership bid, although Ms Eagle and her team are thought to be hoping to persuade him to "back off".
Former home secretary Alan Johnson, who led the Labour In campaign in the EU referendum, added to the pressure on Mr Corbyn with a scathing denunciation of his performance.
In a letter to his local constituency party, he accused the Labour leader of an "inability to take responsibility, demonstrate leadership or give the slightest indication that he is capable of moving beyond meaningless platitudes".
Mr Watson blamed Shadow Chancellor John McDonnell for refusing to allow Mr Corbyn to resign, saying he had tried to get a "negotiated settlement but he was unwilling to move from the position he is in. We are still in an impasse".
"He has obviously been told to stay by his close ally John McDonnell. They are a team and they have decided they are going to tough this out. So it looks like the Labour Party is heading for some kind of contested election."
Mr McDonnell dismissed the claim as "ludicrous" and indicated that he was confident Mr Corbyn would see off any challenge to his leadership.
Addressing students at the School of Oriental and African Studies in London, he bitterly condemned the MPs trying to oust Mr Corbyn.
"It was like a lynch mob without the rope," he said. "If Jeremy had walked on water during the (referendum) campaign he would have been blamed for the loss. What we are watching is a leadership coup."
The fear for the rebels seeking to oust Mr Corbyn is that the party members who elected him by a landslide last year will rally behind him again, leaving him even more firmly entrenched as Labour leader.
There has been speculation about whether Mr Corbyn would have the automatic right to stand under Labour Party rules or whether he will also need to secure 51 nominations from MPs and MEPs in order to get on the ballot paper.