The party leadership is instead expected to order its MPs to abstain during the vote, which will take place when MPs debate a potential delay to Brexit tonight.
The motion, tabled by The Independent Group's Sarah Wollaston, demands that Theresa May delay Brexit “for the purposes of legislating for and conducting a public vote in which the people of the United Kingdom may give their consent" for either leaving the EU on the terms of a deal agreed by Parliament or remaining in the bloc.
It will be the first time that MPs have voted on whether to call another public vote and comes on another day of high drama in Westminster.
Ms Wollaston's motion is backed MPs from Labour, the SNP, Plaid Cymru, the Liberal Democrats and The Independent Group.
But the Labour leadership said it had come at the wrong time. The party has said it supports a public vote to prevent "a damaging Tory Brexit" or a no-deal outcome, and did not rule out backing a motion on another referendum at a later date.
Speaking in the Commons, Sir Keir Starmer, the shadow Brexit secretary said: "Today is about the question of whether Article 50 should be extended and whether we can find a purpose.
"Many colleagues in and out of this place absolutely supportive of the cause of a People's Vote vehemently disagree with this amendment being tabled and voted on today."
He added: "Those pressing this amendment seem to be out of step with the vast majority of co-campaigners who are campaigning for exactly the same cause.
"They may genuinely have a difference of opinion but we will not be supporting [amendment] H tonight."
His announcement was met with cries of "shame on you" from The Independent Group MPs.
But Sir Keir said the People's Vote campaign had also said it was not the right moment to force a Commons vote on another referendum.
A spokesperson for the campaign said: “The People’s Vote campaign does not instruct its supporters in parliament on how to vote. We recognise there is a range of opinions on when to press the case for the public being given the final say, which means some of these MPs will vote for the Wollaston amendment, some may vote against, and some will abstain.
“But we do not think today is the right time to test the will of the House on the case for a new public vote. Instead, this is the time for parliament to declare it wants an extension of Article 50 so that, after two-and-a-half years of vexed negotiations, our political leaders can finally decide on what Brexit means."
The Best for Britain campaign, which is also campaign for another referendum, took a similar position. A spokesperson said: "We believe bringing this issue before parliament today will not truly test the will of the House for a public vote.
"The debate today must focus on the pressing need for an extension to Article 50 to confirm parliament’s rejection of a no deal cliff-edge on 29 MNarch and protect the country from a damaging disorderly exit."