The party has tabled a Commons motion that would give parliament the chance to debate and vote on alternatives to Theresa May's deal.
It said these should include a softer Brexit and a public vote on whatever option parliament decides to pursue.
Labour's motion is one of several alternative proposals tabled by MPs ahead of next week's vote on Ms May's Brexit plan.
The prime minister is expected to try to persuade the EU to make changes to the controversial Northern Ireland backstop after her deal was overwhelmingly rejected by MPs last week.
The Labour proposal is unlikely to pass, with Conservative supporters of a fresh referendum understood to believe the motion confuses matters. They are likely to back alternative motions that simply call for a public vote.
Jeremy Corbyn has so far refused to agree to MPs' and party members' demands that he back a Final Say vote, but said the option should be considered by MPs.
He said: "The prime minister is both refusing to change her red lines or take the threat of a no deal exit off the table. MPs must now act to break the deadlock.
"Our amendment will allow MPs to vote on options to end this Brexit deadlock and prevent the chaos of a no deal.
"It is time for Labour's alternative plan to take centre stage, while keeping all options on the table, including the option of a public vote."
Labour supporters of a fresh referendum welcomed the party's motion. Ben Bradshaw called it "important progress" and David Lammy tweeted: "This is a big step foward. I hope this amendment can be the beginning of the end of years of argument over Berxit in this country."
But while more than 70 Labour MPs have publicly called for another referendum, dozens of others are strongly opposed, including shadow ministers who would likely resign in order to vote against any proposal for going back to the public.
Soon after the party's motion was tabled, shadow housing minister Melanie Onn announced that should would step down if she is ordered to vote for another referendum.
In a letter to constituents, she wrote: "I am not in favour of a People's Vote and will not vote for it. If it comes before parliament as it is expected to in the next few days/weeks, I will instead vote for a better 'Labour' deal.
"This may mean that I am required to step down from my frontbench housing role if it is a whipped vote. I would be very disappointed if this were the case, as I am very proud of the work I have been doing alongside John Healey and Sarah Jones, but recognise that I cannot vote against the frontbench and remain a part of it."
Rebecca Long-Bailey, Labour's shadow business secretary, said the amendment would simply "widen the debate and put all options on the table to avoid a no-deal scenario" but that the party's "priority" remained securning a different Brexit deal, including a customs union with the EU and a closer relationship with the single market.
Asked if Labour's motion makes a second referendum more likely, she told Sky News: "Well, no. In line with our conference policy...our priority is to secure a deal but obviously all options need to remain on the table and that includes a public vote."
She added: "There's no guarantee that it will pass even if our amendment passed and certainly right across the Conservative Party and in the Labour Party there's divided opinion as to whether a second referendum should be a preferred option at this stage."
Ms Long-Bailey suggested that, even if its amendment passes, Labour would not necessarily instruct its MPs to vote for another referendum, saying: "That's something that the party would need to decide at the time."
With parliament still deadlocked over Brexit, former Conservative leader William Hague said a fresh public vote was now "quite likely".
Writing in the Daily Telegraph, he said: "It will probably produce only a narrow victory for either side despite a long struggle, and leave a wounded country to limp on afterwards.
"But it's coming, unless some extraordinary turn of events prevents it. It's coming by a process of elimination, not because it's a good idea in itself."
MPs will vote on 29 January on amendments to Ms May's proposed plan.
Commons home affairs committee chair Yvette Cooper has tabled a motion that would delay Brexit until the end of the year if no deal is in place by the end of February. With significant cross-party support, the motion is likely to pass if Labour agrees to support it.
Hilary Benn, chair of the Brexit commitee, has put down a separate amendment calling for MPs to hold a series of "indicative votes" on various Brexit outcomes.
And Dominic Grieve, the former attorney general, is expected to propose giving parliament to right to seize control of Brexit if the government cannot pass its deal.
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