"What people are trying to do at the moment ... is to try to come up with a negotiated solution to the problem that we have and a way forward, and I don't want to compromise that because, as I said, this isn't about me", Umunna said.
MP Helen Goodman has joined other Labour MP rebels, saying that Mr Corbyn can not set out a "clear alternative for the country".
Mr Corbyn is still hanging on at the top of the Labour party despite growing pressure to resign from his own MPs.
He added: "I will continue to uphold the democratic wish of the members of the party, and respect their decision for Jeremy Corbyn to continue his role as the leader".
It was set up in an attempt to win backing for a centrist candidate and replicate the left's Labour Party recruitment drive which helped propel Corbyn to victory in last summer's leadership election.
Jeremy Corbyn's leadership of the Labour Party was backed by a major union - but it stressed that the party must function as an "effective opposition".
A defiant Corbyn also made it clear that he would again be a candidate if a leadership election were triggered.
"Therefore, for the sake of the Party but more importantly, for the sake of those who need a Labour Government, Jeremy needs to resign".
"Our priority must now be to mobilise this astonishing new force in politics, and ensure people in Britain have a real political alternative".
Mr Corbyn's supporters are buoyed by the huge increase in Labour Party membership.
The Pontypridd MP said: "Over the last week, I have been contacted by hundreds of Labour members and MPs deeply anxious that the Labour Party is truly in danger of splitting apart".
Mr Corbyn has so far resisted calls to step down as leader after MPs voted by 172-40 that they had no confidence in him.
Former shadow business secretary Angela Eagle has insisted she is ready to challenge Mr Corbyn if he refuses to stand down while Owen Smith, the ex-shadow work and pensions secretary, has said he is "ready to do anything I can to save and serve the party" amid speculation he was also considering a leadership bid.
An overwhelming majority of Labour MPs passed a vote of no confidence in Corbyn.
Kinnock went on to say: "It is vital, essential, irreplaceable that the leader has substantial support from those who go to the country and seek to become lawmakers".