Jeremy Corbyn is facing grassroots pressure for Labour to come out firmly in favour of a second Brexit referendum.
The campaign group Another Europe is Possible has said that unless Labour opposes EU withdrawal, it will lose millions of votes.
The group said that 201 constituency Labour parties are set to debate a motion calling for a stronger line against Brexit by the end of this month.
As well as calling for opposition to EU withdrawal and a new referendum, the Another Europe is Possible-drafted motion urges a “principled stand in defence of free movement and for democratisation of European institutions”.
The move comes ahead of a meeting of the Labour Party’s International Policy Commission on Wednesday, which is set to discuss Brexit.
Ana Oppenheim, an organiser for Another Europe is Possible, said: “As the sequence of events unfolds in Parliament, we need to look a the reality. Brexit is an attack on working class people, dripping with imperial nostalgia and migrant-bashing.
“If Labour fails to oppose it, we will lose millions of voters, and if it happens, any attempt to deliver a socialist programme will be hamstrung from day one.
“By the end of January, we reckon that a majority of constituency Labour parties could have debated a motion – and a big majority will pass it.”
A recent study of the attitudes of more than 1,000 Labour members found that 72% want Mr Corbyn to throw his weight behind a new so-called People’s Vote on EU withdrawal.
Mr Corbyn has said that Labour’s preferred option to deal with the Brexit situation is to force a general election, but the Labour leader has insisted that all options remain on the table.