Labour leader Jeremy Corbyn has been warned to give clear backing to a second EU referendum, or risk losing millions of pro-EU voters.
The party is split on what to do about Brexit, as are the Conservatives.
Richard Corbett, who leads Labour's MEPs in Brussels, has spoken to the Observer newspaper.
Mr Corbett warned that voters could desert the party in the EU election unless it committed to a public vote.
He told The Observer: "If Labour does not re-confirm its support for a confirmatory public vote on any Brexit deal in its manifesto then it will haemorrhage votes to parties who do have a clear message. If on the other hand, we do offer clarity and a confirmatory ballot we could do very well."
Former Labour foreign secretary Margaret Beckett echoed his calls, saying a "lack of clarity" could cost the party not only in the EU election, but in the next general election too.
She told the paper: "It is very important that there is a clear message about where Labour stands and what Labour is offering. In my view that clear and simple message should be that there should be a confirmatory vote of the British people."
The SNP's Westminster leader, Ian Blackford has said that Mr Corbyn must stop wasting time and commit to supporting another vote on Brexit.
Mr Blackford said it would be "unforgivable" for Labour to avoid another referendum on EU membership, and he warned against pushing through a Brexit deal that ends freedom of movement and removes single market access.
Earlier this week, the EU granted a further extension to Article 50 and the UK is now set to depart on October 31.
Mr Blackford said: "Talks have been ongoing between both the Tory Government and Labour Party for over a week but not a single compromise has been tabled.
"Instead of wasting more time, Labour must make clear to the Tories that the answer to the Brexit deadlock is to put the decision back to the people.
"It is no secret that the Labour leader has ducked and dived on the issue of a second EU referendum, but at this crucial crossroads it is high time Jeremy Corbyn commits to putting a second referendum on the table as the only way forward.
"In 2016, Labour backed Remain and the Tories' Brexit policy has failed - so there is no mandate for any outcome they may reach if it is not put back to the people."
"It would be unforgivable if Labour side-stepped a second EU referendum in the talks and instead pushed through a deal that would drag Scotland out of the single market and end freedom of movement - which will hit our public services and cost thousands of jobs.
"Westminster is failing Scotland and it is becoming clearer each day that the only real way to safeguard our interests is through independence."