As Conservatives battle it out for the leadership of their party, former shadow business secretary Angela Eagle has said she has the support to mount a challenge on Jeremy Corbyn.
But shadow chancellor John McDonnell said the Labour leader had an "open door" to see individual MPs, including Mr Watson.
Mr Corbyn has faced a turbulent week with many shadow cabinet members resigning and 172 MPs backing a no confidence motion in his leadership.
Corbyn has refused to back down despite his own MPs turning against him, and has repeatedly said it is his duty to represent the 59% of Labour members and trade unionists who voted for him in his landslide leadership election past year.
On Monday Mr Corbyn will be quizzed by MPs over anti-Semitism in the Labour Party after stoking controversy at the launch of a report into the issue last week.
Schneider insisted Corbyn would win any fresh contest and insisted the wave of people signing up for Labour membership was mainly supportive of the leader.
Despite speculation, no Labour MP has yet confirmed they will stand against Mr Corbyn in a leadership contest.
Talking about the large numbers of people who paid £3 to become registered supporters of Labour to back Corbyn in last year's election, Kinnock said they were "very welcome in the party" arguing that they brought energy and vibrancy.
On Wedesday, Jeremy Corbyn was caught on camera having a laugh with Marc Wadsworth from the far-Left organization Momentum - who accused Labour MP Ruth Smeeth of "colluding" with the right-wing press during a conference announcing the results of an inquiry into allegations of antisemitism in the party.
Mr Corbyn is still in post a week after the vast majority of his shadow cabinet and frontbench team resigned in an effort to force him out.
"Only last week some elements of the coup plotters were briefing the media that I was trying to supplant Jeremy, that I was launching a coup against Jeremy".
Lord Prescott, touted as a mediator to resolve his party's leadership problems, told the same programme: "I hope Angela and Owen don't go into an election because that will take the fight much nearer to a kind of civil war".
But now more than 500 councillors have added their voices to the criticism of Mr Corbyn.
He said that he is ready to "reach out" to his critics, but if there were a leadership contest he would stand again.
Speaking outside her home, Eagle said: "It's a week since Jeremy lost that vote of no confidence".
"I'd rather we just got on and settled down, to be frank, and got on with our job because the country needs us".
Lord Neil Kinnock said Labour MPs were "doing the clear and honest thing when they are so fundamentally alarmed by the prospects of the party". "I don't care what candidate comes forward, we'll carry on supporting Corbyn".