split between holding a referendum on any deal; holding one with caveats; or rejecting the idea altogether.
The party is also holding talks with ministers to try to agree a Brexit deal and break the deadlock in Parliament.
The EU has set a new deadline of 31 October for the UK's departure.
But Theresa May has said the UK could leave earlier - and avoid European elections - if MPs can agree a deal before 23 May.
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Labour agreed a policy at its last conference that if Parliament voted down the government's deal or talks ended in no-deal, there should be a general election.
But if it cannot force one, it added, the party "must support all options remaining on the table, including campaigning for a public vote".
The party's deputy leader, Tom Watson, has urged members to contact the NEC and called on them to back a "confirmatory referendum" on any deal.
But shadow international trade secretary Barry Gardiner said it would be a change in Labour's policy, which is to "try to deliver on what people voted for".
'A debate going on'
Shadow Brexit Secretary, Sir Keir Starmer, said Labour's frontbench team would meet ahead of the NEC meeting.
"There's obviously a debate going on," he said.
"Shadow cabinet will look at it and then the NEC will look at it... so we'll have to let that process proceed."
After a meeting of the Parliamentary Labour Party on Monday night, MP Neil Coyle - who backs a public vote - said he had lost 500 members in his own constituency over the issue.
He said the NEC should "stand up for members or expect members to leave", claiming they could join Change UK - The Independent Group instead, which is campaigning for a People's Vote.
Labour's shadow attorney general Shami Chakrabarti said it was for Labour frontbenchers involved in the Brexit talks with government to tell the NEC whether there was "any hope now of a sensible, jobs-first Brexit deal" or whether "the only means of breaking the deadlock is a confirmatory vote".
The latest talks between the government and Labour on Monday were described as "positive" and "productive" by the PM's de facto deputy.
Speaking afterwards, David Lidington said he was "encouraged" by a sense from both sides about the "need to inject greater urgency" into the talks.
Labour's shadow environment secretary Sue Hayman said there had been "really constructive discussion", with the two parties "getting much more into the nuts and bolts of the detail", and said that she believed the government was "open to moving forward in our direction".
Cross-party negotiations have been taking place for a number of weeks after Theresa May's Brexit deal with the EU was effectively rejected for a third time by MPs.
Mr Lidington said there would be further meetings between the parties this week.