"But there is a way through this where actually MPs who stood at the general election saying they would just get on with Brexit - just get on with Brexit!"
He has previously showed his preference for a new national election, nearly three years since Britain voted to leave the European Union which left both his party and the governing Conservatives deeply split over how, when and whether Brexit should happen.
Some senior figures, though, still want him to go further - to campaign for another referendum now and wholeheartedly fight to remain.
This week's guest is Richard Corbett, leader of the UK's Labour contingent in the European Parliament.
Gary Hynds, deputy chairman on political issues for the Northern Ireland Conservatives, said the poll results don't represent what he is hearing in the party, although he admitted that "it is disappointing that colleagues think that way".
Some are also anxious about appearing to cave in to Watson, whom they believe is using Brexit as a "wedge" issue, to undermine Labour members' support for Corbyn's leadership.
"The Peterborough by-election result, with Labour's vote share down 17 per cent and the Brexit Party coming so close, gives a stark warning of what could happen in Tory-Labour marginals, the majority of which are Leave seats".More news: Former XFL Icon Rod Smart 'He Hate Me' Found Safe
More news: Earthquake in southern China kills 11 people, injures 122
More news: Deputy in clash with Raptors president Masai Ujiri considering lawsuit, says lawyer
Addressing his top policy team, Corbyn said on Wednesday that any such public vote should offer "real choices for both Leave and Remain voters".
The outcome of today's meeting is therefore not the adoption of a new position on Brexit, but it formalises the policy settled on by the Labour leader last month. Other Labour party personalities have expressed similar opinions, like Emily Thornberry and Sir Keir Starmer.
Corbyn, who led Labour to a much stronger than expected showing in a 2017 general election, has long wanted to move the conversation away from Brexit, seeing his anti-austerity message as a vote victor.
In his opening remarks to Tuesday's shadow cabinet meeting, he told his frontbench team Labour's evolving position on a referendum was "in line" with the stance agreed by party members a year ago that a public vote was "an option" if other avenues to prevent a no-deal exit were exhausted.
One thing Labour's divided shadow cabinet does agree on, I'm told, is a reluctance to put "No Deal" on the ballot paper in any public vote, because of the "chaos" leaving without a deal would course, particularly on security.
The statement essentially confirms a shift in the Labour Party's position toward a staunch "remain" platform, as a new vote would be a huge blow to victorious leave voters.