Dozens of Labour MPs might be prepared to go against the party’s leadership if there is a vote on starting the Brexit process, the BBC understands.
Jeremy Corbyn has said all his MPs will be told to approve the triggering of Article 50 because they should accept the result of last year’s referendum.
Lib Dem Tim Farron says generations to come will not forgive that position.
The Supreme Court will announce next Tuesday whether the government needs to seek Parliament’s approval.
Ministers say they already have enough powers under the Royal Prerogative to go ahead with Brexit.
But campaigners argue that starting Brexit in this way would be undemocratic and unconstitutional.
In June’s referendum, 51.9% of voters backed leaving the EU, while 48.1% supported remaining in the 28-nation group.
Mr Corbyn said he did not think it was right to block Article 50 in the wake of the referendum result.
“It’s up to us to use the opportunity that’s provided to stop the Tories from doing this bargain basement, low tax haven on the shores of Europe,” he told the BBC.
“What I’m saying to all of my MPs is we’ve supported the principle of holding the referendum, the referendum was held, it delivered a result – I don’t think it’s right to block Article 50 negotiations.
“It’s absolutely right that we’re involved in these negotiations and making the case for a fairer and socially just Britain.”
Asked if that meant he would be imposing a three-line whip – the strongest available sanction – on Labour MPs, requiring them to back Article 50, he said: “It means that all Labour MPs will be asked to vote in that direction next week or whenever the vote comes up.”
However, a senior Labour source has told the BBC between 60 and 80 of the party’s MPs might be ready to defy the leadership if there is a vote in Parliament.
Shadow business secretary Clive Lewis said the government’s current position is “unacceptable” and he wants further assurances.
Another Labour MP told the BBC there would be a “swathe” of resignations from the front bench if Mr Corbyn instructed his MPs to vote for Brexit.
The MP said that for colleagues in constituencies that voted strongly for Remain it would be “suicide” to back Article 50.
And Labour’s Mike Gapes, who has been outspoken in his criticism of Mr Corbyn in the past, told the Ilford Recorder he would not be toeing the party line.
“I am going to vote against triggering Article 50 and let me be clear, I am going to be as loyal to Jeremy Corbyn as he was to previous Labour leaders,” he said.
“I will show the same loyalty he did when he voted, 500 times, against the Labour whip under successive party leaders. The people of Britain did not vote to become poorer and I will not vote in favour of any deal that would see us leave the single market.”
BBC political correspondent Ben Wright said nobody expected Parliament to stop Brexit being triggered, but Mr Corbyn could struggle to keep his party’s position coherent, whether he insists on a three-line whip or not.
Lib Dem leader Mr Farron accused Labour of “lamely” giving up against the government’s drive for a hard Brexit.
He said he believed Mr Corbyn had put the party on the wrong side of the biggest political issue in a generation.
“I think what Labour has done is to believe this is too difficult for them politically, let’s just wait for it to go away, and the meeker we are, the quicker it will go away,” he told the Guardian.
“I think that’s the calculation they’ve made, and this and future generations are not going to forgive them for that.
“It’s not divisive to hold the government to account, and not just to lamely give up as we go over a cliff, and that is what Labour are doing – they are being the most ineffective opposition in living memory.”
Mr Farron added that his party, which has just nine MPs, would not consider an electoral deal with Labour because Mr Corbyn is “electorally toxic”.
Green Party co-leader Caroline Lucas claimed Mr Corbyn was “trying to deny Labour MPs the chance to make their own principled choice on one of the most important decisions of the UK’s recent history”.