Sky's Chief Political Correspondent Jon Craig said the deputy leader told Labour MPs the trade union meeting was "the last throw of the dice" after they overwhelmingly backed a vote of no confidence in Mr Corbyn last week.
"Since my election as leader of our party a year ago, we've won every parliamentary by-election, some with vastly increased majorities".
But she said: "The country doesn't have an effective opposition at the moment".
"If there is a leadership election then people are entitled to back any candidate they wish".
On Thursday, former shadow cabinet members Angela Eagle and Owen Smith again called on Mr Corbyn to quit.
But Annie Thomas, a local Momentum member, said the room the meeting was held in was not big enough, a large amount of people had turned up and it had been hot, "so you could feel the tension building" and she said there was some "heckling from both sides".
However, a confessed "proverbial optimist", Zeichner retains a positive outlook for the Labour Party, saying that the party's current difficulties were caused by a "psychodrama in the Conservative Party" and that "Labour will emerge from this much more strongly", since "the divisions in the country can not be solved by a Conservative market approach".
Ms Eagle, who was expected to launch her bid for the top job last week, warned that the threat of a contest had not gone away. Yesterday, I spoke directly with Len McCluskey of Unite and met with our leader, Jeremy Corbyn, to explore what I can do to try and heal the rifts that have opened up in our movement.
"We'll be sending a letter to Mr Corbyn pledging our support, and we'd like him to come to Keighley to meet people here".
After the Labour parliamentary party passed a no-confidence motion in Corbyn, talks to encourage him to resign have remained inconclusive.
One passionate young demonstrator shouted into a microphone, "I want Jeremy Corbyn as leader more than anything in the world".
As Mr Corbyn released the video, it emerged that another frontbencher has resigned from his position.
One Labour source said: "Alan Johnson has been one of Mr Corbyn's most severe critics so this vote will be seen as a blow to his authority".
The prominent backbencher Stephen Kinnock, the MP for Aberavon, said the referendum result meant Labour needed "a persuader, not a protester", to hold the Conservative government to account through the tough negotiations with the rest of the European Union that will lie ahead.