Former shadow business secretary Angela Eagle - who quit in the mass walkout of frontbenchers - appears ready to announce she will run as a “unity” candidate, while former shadow work and pensions secretary Owen Smith is also reportedly considering a challenge.
Mr Watson warned the party was “in peril” after Mr Corbyn refused to discuss some form of “negotiated settlement” following Tuesday’s unprecedented vote of no confidence by Labour MPs.
He 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.”
The big 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 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 yelling: “What about Europe? Where were you when we needed you?”
Mr Watson’s intervention came after former leaders Ed Miliband, Gordon Brown and Harriet Harman joined calls for Mr Corbyn to quit.
Former home secretary Alan Johnson, who led the Labour In campaign in the referendum, added to the pressure with a scathing denunciation of Mr Corbyn’s 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”.
The Labour leader, however, remained defiant, with a spokesman saying: “Jeremy Corbyn is determined to carry on with the job he was democratically elected to do.”
Mr Watson blamed hard left shadow chancellor John McDonnell for refusing to allow Mr Corbyn to resign.
“I went to see Jeremy today to see whether we could find a way of getting a negotiated settlement but he was unwilling to move from the position he is in. We are still in an impasse,” he said.
“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 that Mr Corbyn would see off any challenge to his leadership.
Addressing students at the School of African and Oriental Studies in London, he bitterly condemned that MPs trying outs 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 grassroots activists who propelled him to the leadership last year will return him again in any new contest, leaving him even more firmly entrenched in his position.
The general secretary of the Fire Brigades Union warned Labour MPs calling for Mr Corbyn to resign that they are selected as candidates by local party members.
Matt Wrack told Good Morning Britain: “How do people become Labour MPs? They become Labour MPs because local Labour Party members decide that they should be the candidate.
“They are there because they are Labour Party members and they are selected locally. The members have an important role in this.”
He added: “There’s not a single MP in Parliament from the Labour Party that has done it because they are Angela Eagle or Hilary Benn, or whoever, or Jeremy Corbyn.
Corbyn loyalist Barry Gardiner - promoted to the shadow energy secretary role after the mass resignations - said MPs had to recognise the PLP “does not own the Labour Party”.
He told BBC Radio 4’s Today programme that the vote had changed little as it was clear the leader had “never had” the confidence of many Westminster colleagues and accused critics of “letting the Government off the hook”.
Some MPs suggested Mr Corbyn was willing to quit but was being kept in place by allies and advisers on the left of the party.
Barry Sheerman tweeted: “Sure Jeremy Corbyn would have decided to resign but it appears John McDonnell has instructed him to stay put.
“Not a person whose judgement I have ever been comfortable with.”