He appealed to the party’s MPs to “calm down” and settle Labour’s differences through a democratic process.
Answering questions following a speech in London he said Mr Corbyn is “staying as the leader of the Labour Party” and if there was a contest he could win.
He said it was “really disappointing” that frontbenchers had quit at a time when the country needed stability.
The Labour leader has vowed to stay on despite the threat of a challenge and the mass resignation of scores of frontbenchers in protest at his leadership.
Shadow cabinet loyalist Emily Thornberry hit out at the “irresponsible” MPs plotting to oust Mr Corbyn and insisted the party was electable with him at its head.
Mr Corbyn has repeatedly said he will not quit because he won an overwhelming mandate from 60% of Labour members in the 2015 leadership election and stepping down would “betray them”.
But a YouGov poll for The Times suggested that although he might win a leadership contest, his support had fallen away since May.
The survey, carried out between Monday and Thursday as Mr Corbyn struggled to maintain his position, found 50% of Labour members surveyed would vote for Mr Corbyn but 47% would not - down from the 64%-33% lead in May.
Angela Eagle, who was the most senior member of the shadow cabinet to quit, is delaying her expected leadership challenge to Mr Corbyn.
The former shadow business secretary had been expected to declare that she was going to run as a “unity candidate” at a news conference on Thursday afternoon.
The YouGov poll indicated that Mr Corbyn would beat Ms Eagle by 50% to 40% if they went head to head.
Despite the mayhem within the Labour ranks in Westminster, 51% of the party’s members believed Mr Corbyn was doing well, with 48% saying he was doing badly.
Some 44% said he should step down now as leader of the Labour Party, according to the study, but 60% said he should lead the party into the next general election.
Shadow foreign secretary Ms Thornberry told BBC One’s Question Time that after the Brexit vote “people have, instead of thinking about the jobs of people in Britain, have been thinking about David Cameron’s job and Jeremy Corbyn’s job and whether they can get them or not”.
“I personally think that is irresponsible at a time when our country needs us.”
She added: “It is incumbent on us to have some cool heads and to think through what is the best for the country - and what is the best for the country means a united opposition that can speak clearly.”
With the prospect of Mr Corbyn facing a leadership challenge, around 60,000 new members have joined Labour in the last week, according to party sources.
Both factions in the party have been active on social media trying to recruit members to vote in a contest to either support or oppose Mr Corbyn.
Despite the party meltdown, Mr Corbyn attempted to show it was business as usual by holding a press conference on the findings of an inquiry into anti-Semitism on Thursday.
But the event did little to strengthen his beleaguered leadership, fuelling calls for him to quit and sparking anger among the Jewish community.
Labour’s Ruth Smeeth called on the leader to stand aside to make way for “someone with the backbone to confront racism and anti-Semitism” after he failed to stand up for her when one of his grassroots supporters launched a verbal assault on her during a question and answer session.
Ms Thornberry said anti-Semitism was not “rampant within the Labour Party but we ought to hold ourselves to a higher standard than the rest of the country and politicians in our Labour Party should hold ourselves even at a higher standard than that”.
Ed Miliband’s former advisor, Tom Baldwin, said Mr Corbyn needs to start thinking about what is right for the party.
“He does not have the support of MPs, he does not have the support of MEPs, he does not have the support of Labour councillors,” added Mr Baldwin speaking on ITV’s Good Morning Britain.
Referencing the poll Mr Baldwin said that is why this “impasse” has been reached.
“He cannot lead the party or MPs anymore, but if he goes into a leadership election he may well be re-elected by those members,” he said.
“In these circumstances a responsible leader, someone who has the party’s interests at heart, has to recognise he can no longer lead.”
Mr Baldwin said Mr Corbyn appears to be “agonising” about what the right thing to do is, and is surrounded by one or two who are “desperate for him to cling on” in a bid to keep control of the party.