The grassroots group Momentum - set up to support Jeremy Corbyn's leadership of the Labour party - looks set to begin a consultation of its members on its Brexit stance, ahead of the party's conference in Liverpool next month.
Momentum's ruling body will meet next week and will consider a petition, signed by more than 4,000 members, calling for the organisation to canvas opinion in its ranks, including on whether to call for a conference vote on holding a referendum on the final Brexit deal.
Support for the idea has been growing among Labour members - and backing from an influential group like Momentum would increase pressure on the party leadership.
The Shadow Brexit Secretary Keir Starmer's comments that "all options should be on the table" and the Shadow Chancellor John McDonnell's assertion that Labour was "not opposed to any form of democratic engagement" - have increased speculation that the party could be considering a shift in position.
Any consultation of Momentum members would have to be approved by its National Co-ordinating Group, which will meet on 1 September.
"If we were staring down the barrel of a no-deal gun, I personally would like to be asked what I thought about that," said Laura Parker, Momentum's National Co-ordinator.
"Because however you voted, you didn't vote for this country to throw itself off a cliff...I could see a lot of Momentum members being very concerned."
A number of Remainer Labour MPs, like Chuka Umunna and Chris Leslie, are openly calling for a new referendum and have allied with the so-called People's Vote Campaign, which also has the Liberal Democrats and a few Conservatives on board.
But - significantly, there's now a growing clamour from Labour members on the left of the party, who form Jeremy Corbyn's base.
The group Labour for a People's Vote are trying to jump through the hoops of the party's complex internal policy procedures to make sure a public vote on the final deal is discussed in Liverpool.
Michael Chessum, from the group, says they've drawn up motions to submit to Conference - backing another referendum - which are being considered by more than 200 constituency labour parties and have been passed by at least 13, with dozens more meetings still to take place.
"We are not sitting around and waiting for white smoke to come from anyone's office and we're not waiting for chess games to play out in Westminster," he says.
"We are using the democratic levers of the labour movement and we are pushing for a vote at conference."
Making it over all the procedural hurdles to get the issue to that stage will be difficult - but backing from Momentum and its delegates would certainly help, as would support from the unions.
Len McCluskey, the leader of the biggest trade union Unite, isn't backing the idea outright but has said "we leave open the option of a people's vote on the deal".
At the moment though, Keir Starmer is clear about the sequence of events.
"Our focus has been on a meaningful vote in parliament, " he says. "We have set out six tests for the prime minister and we will faithfully apply those tests to the deal that comes back.
"If they're not met, we will vote against that deal. And if parliament as a whole votes against that deal, our view is that no deal is not the inevitable consequence, something else must happen, parliament must decide and all options should be on the table."
It is worth remembering that Parliament would need to pass legislation for another referendum to be held, and the government has ruled out the possibility.
Never mind the practical issues that would arise of the exact question and the timing.
But as the Brexit negotiations enter their final phase and with the cabinet and parliament still divided, it's unlikely that calls for a fresh vote on the final deal will die down.