Shadow chancellor John McDonnell has signalled Labour will whip its MPs to back a bid for a new Brexit referendum.
Labour divisions on the issue were thrown into sharp focus as former minister Caroline Flint estimated up to 70 of the party's MPs are opposed to a second national EU withdrawal poll.
Mr McDonnell told Sky News' Sophy Ridge on Sunday: "Normally we will whip and that will be decided in the normal way by the chief whip and the shadow cabinet and the party overall.
"I think on an issue as this we would see a whip but also you've got to respect people's views and their constituency interests as well, and the whipping arrangement will be determined in discussion in due course.
"I just say this – and I think it'll be for MPs right the way across the House in all parties now – that they've got to look to the long-term interests of the country, they've got to protect people's jobs, they've got to protect the economy, otherwise we'll never be forgiven in the future."
Ms Flint appealed for the party leadership to allow a free vote and also urged colleagues to back an improved Brexit deal given the 2017 election promise to respect the 2016 referendum result.
The MP told Sky News: "My appeal to John McDonnell, to Jeremy Corbyn, to Keir Starmer, is allow MPs to have a free vote on an improved deal.
"So those MPs who want a second referendum can vote for that but those of us who want to keep our promises to our electorate can also keep faith with those people and vote for an improved deal."
Ms Flint also said: "I think there is something like 60 or 70 Labour MPs who feel as strongly as I do against a second referendum."
She added if there was a free vote among Labour MPs then "tens, twenties, thirties would vote for an improved offer".
MPs have been told by Prime Minister Theresa May they will have a "meaningful vote" on her Brexit plans by March 12, and Labour is expected to use the occasion to put down an amendment on a second referendum.
The PM said that if her deal is rejected, MPs will be able to vote on whether the UK can leave the EU in a no-deal scenario, and if that is rejected, the Commons can decide on whether to extend Article 50 and delay Brexit.
In a message to Labour MPs on Brexit, former prime minister Tony Blair told BBC1's Andrew Marr Show: "Vote against the deal, use an extension to come to a conclusion – hard versus soft or back to the people.
"I think you'll get to another referendum when people understand that a hard Brexit is going to be deeply economically painful for the country and a soft Brexit means we just become a rule-taker.
"It's in those circumstances that I think you mobilise a majority in Parliament to say the sensible thing in these circumstances is to put it back to the people, or pass her deal subject to a confirmatory referendum."
Asked if delaying departure from the EU beyond March 29 would be going back on a promise, International Trade Secretary Liam Fox told BBC1's The Andrew Marr Show: "It would be very unfortunate were that to happen.
"But, if we have no option, in order to deliver a smooth Brexit, then so be it."
Pressed on whether he would choose a no-deal exit or an extension of Article 50 if the PM's Withdrawal Agreement is defeated again, Justice Minister Rory Stewart said: "Personally, I think we would have to be forced into an extension for Article 50."
The comments came after Mrs May was given a Brexit boost as key Tories signalled swinging behind her stance in exchange for movement on the Northern Irish backstop.
Chairman of the highly influential 1922 Committee of backbench Tories Sir Graham Brady expressed optimism that a breakthrough on the backstop was close.
The hardline European Research Group (ERG) of Tory MPs led by Jacob Rees-Mogg also indicated a more conciliatory tone on the issue.
The ERG has drawn up "three tests" the Government must pass to win backing, according to the Sunday Times.
"If there's a choice between no deal and an improved deal then we should seriously consider an improved deal" says Labour MP Caroline Flint. #Ridge
— Ridge on Sunday (@RidgeOnSunday) March 3, 2019
In private talks with Attorney General Geoffrey Cox, the ERG called for a legally-binding mechanism to escape the backstop, with a clear exit route and an unambiguous rewrite of the language in the Government's legal advice, the newspaper said.
The stance has been drawn up in conjunction with the DUP, according to the Sunday Times.
Sir Graham made clear he could swing behind the Prime Minister's Withdrawal Agreement ahead of crunch Commons votes.
Writing in the Mail on Sunday, Sir Graham said: "The whole country is tired of vacillation and delay.
"When the right compromise is offered, we should pull together behind the Prime Minister and help her to deliver our exit from the European Union on March 29."
Many Brexiteers have expressed opposition to the backstop as it would leave the UK obeying EU customs rules if no wider trade deal is agreed after a transition period.