Umunna had previously thrown his hat into the ring for Labour leadership past year, but withdrew quickly to back Liz Kendall, who ended up coming last, behind Corbyn, Andy Burnham and Yvette Cooper.
The influx comes with Mr Corbyn under huge pressure from his rebellious MPs.
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.
Michael Foster, who represented the city from 1997 until 2010, questioned whether Mr Corbyn could lead the party to win a future general election.
The embattled Labour leader has refused to step down despite facing an overwhelming vote of no confidence by Labour MPs and mass resignations in his shadow cabinet, repeatedly citing his popular support at the party's grass roots. The peer said that despite this, he felt that "there's something rather more unpleasant" in Mr. Corbyn's character.
"Angela Eagle - if you look at her constituency - her local constituency passed a motion in support of Jeremy Corbyn".
Mr Corbyn was said to have responded by making clear he had no intention of leaving, and put out a video appeal to supporters to unite behind his leadership.
"Unfortunately, Jeremy Corbyn is not providing the leadership required".
Corbyn underlined the point in an article for the Guardian newspaper on Friday, saying: "MPs need to respect the democracy of our party and the views of Labour's membership, which has increased by more than 100,000 to over half a million".
She added she was called a "traitor and a conspirator" in a meeting which included threats which "felt dangerously close to bullying". One agency source said: "If I was advising a client on this I would tell them there's only so many times you can say something is ridiculous".
On Monday morning Ms Eagle told Sky News: "I have the support to run and resolve this impasse and I will do so if Jeremy doesn't take action soon".
"Everyone says what a nice chap he is, but if he doesn't understand that, I'm beginning to feel that maybe he's not quite such a nice chap".
Nus Ghani, Wealden MP, asked why the Labour Party needed a report telling its members not to use Hitler, Nazim, or Holocaust metaphors, distortions or comparisons, particularly in debates about Israel and Palestine.
"We need to challenge this Tory Government, we need Labour to fight to minimise the damage Brexit is doing to our economy and our society".