The "time has passed" for Labour to be able to get its own Brexit deal and any agreement should go to a vote, shadow cabinet minister Andrew Gwynne says.
He said the party "will do everything we can to stop no deal".
But Caroline Flint, one of 26 Labour MPs who have urged Jeremy Corbyn not to call for a second referendum, said she would prefer no deal to no Brexit.
She said the number of Labour MPs prepared to back the government's Brexit deal "will go up" in the future.
Labour had promised another Brexit vote in certain circumstances, specifically if it could not get its own deal with the EU passed by MPs or if there was no general election.
Following the party's poor performance in the European elections last month, Mr Corbyn said there "had to be a public vote" on any deal agreed with Brussels.
Some senior figures, though, have wanted him to go further - to campaign for another referendum now and wholeheartedly fight to remain.
Others, like Ms Flint, have warned against another Brexit referendum.
Speaking at a shadow cabinet meeting last week, Mr Corbyn said he would be listening to colleagues and consulting with trade unions before officially setting out Labour's position.
"The realisation is that whatever they come back to Parliament with between now and October, we have to go back to the people and we have to say 'is this really the Brexit that you want?" Mr Gwynne told Sky News's Sophy Ridge.
He conceded that it would be "incredibly difficult" to explain to Labour leave voters the case for a second referendum, but added that the party's position was evolving because of the increasing risk of a no-deal outcome.
He said Mr Corbyn had intimated that Labour was now in favour of a second EU referendum in all circumstances "because it's looking increasingly likely that we're heading to no deal, that was not on the ballot paper in 2016".
It comes after a letter from 26 Labour MPs - including Ms Flint - warned Mr Corbyn .
They claim another referendum would be "toxic" and empower the "populist right" in many Labour heartlands.
Former minister Ms Flint told the BBC's Andrew Marr Show that she would be happy to cross party lines and "talk to anybody to stop no-deal".
"The health of the nation needs us to agree a deal and move on," she said.
But she stressed that in the event the UK was heading for a no-deal Brexit, she would not vote to revoke Article 50 - the process by which the UK leaves the EU.
"Those of us who signed this letter, and others, believe that we should respect that promise [to honour the referendum result], and that is what has been Labour party policy and that is what is meant to have guided us through Parliament," she said.
However, shadow international development secretary Dan Carden said he would vote to stay in the EU in a second referendum if no deal was the other option.
He told BBC Radio 5 Live's Pienaar's Politics: "It's a really difficult question to answer when you don't have the Leave option negotiated, so we don't know what the Leave option is.
"If there was a referendum and no deal was an option and Remain was an option, then I would be voting to stay in the EU."