The Unite general secretary, Len McCluskey, has privately told Labour MPs the party should have severe reservations about backing a fresh Brexit referendum, saying voters could see it as a betrayal.

The deep scepticism from one of Jeremy Corbyn’s closest and most powerful supporters is likely to unnerve MPs and campaigners hoping the party is warming to the idea of a fresh Brexit vote.

Labour’s deputy leader, Tom Watson, hit back at McCluskey’s warning, laying bare the tensions in the party.

“To suggest it represents a ‘betrayal’ grossly distorts Labour’s position and is deeply unhelpful to those seeking a solution to an an issue that is reaching crisis proportions,” he told the Guardian.

McCluskey met around two dozen MPs in the House of Commons on Tuesday, where he voiced fears about the impact that backing a second Brexit vote would have on Labour.

The private intervention comes at a fractious time for the party, as the leadership agonises over what its tactics should be if Theresa May loses a vote on the Brexit deal in parliament.

McCluskey is an influential figure in the party and several of Corbyn’s most trusted staff are former Unite members and close to the union leader, including the party’s general secretary, Jennie Formby, the adviser Andrew Murray and the chief of staff, Karie Murphy.

Yet among those around Corbyn there is an intense struggle over whether the party should consider moving towards a “people’s vote”.

MPs from the Unite group in parliament met on Tuesday evening at the invitation of McCluskey. Some agreed with his reservations, a source in the meeting said, with three arguing forcefully in favour of a second vote.

McCluskey said he believed Labour MPs would need to eventually back some version of a Brexit deal, sources said.

One MP in the room said the meeting was informal and a chance for McCluskey to give his “view of the world”, but that they had been struck by the strength of his scepticism about another referendum.

Another source said that while McCluskey argued forcefully against a second referendum, he wanted to hear other people’s views.

“Len spoke about tackling people’s concerns about immigration and the exploitation of immigrant labour. He said there would be a sense of betrayal among the members if we went for a second referendum.

“He said the party has to take concerns about immigration into account, and that if politicians had done so sooner we might have avoided the current situation. I wouldn’t say that majority of MPs agreed with him at all. But he gave clear examples where the exploitation of immigrant labour has undermined existing wage levels.”

A Unite spokesperson said: “Len McCluskey repeated a number of times that a Conservative Brexit would be bad for working people. He urged MPs to hold their nerve as Labour’s alternative approach would deliver for the country.”

Labour’s shadow cabinet has been edging towards backing a second referendum if the party fails to secure its first choice of a general election after the meaningful vote.

The shadow chancellor, John McDonnell, recently suggested it was “inevitable” that Labour would support the idea in those circumstances.

Watson said a second vote would only come after “a failure of parliamentary arithmetic and political leadership”.

“It can only happen if parliament judges that the current government has failed to negotiate a good deal for Britain. That’s why the people’s vote has always been an insurance policy for Labour,” he said.

However, other close allies of Corbyn remain sceptical about how a second vote would go down in leave-backing Labour constituencies; and the party’s pollsters have been researching the views of voters in heartland seats.

There are also fears that a significant minority of Labour MPs would not back the policy, even if it was backed by the frontbench.

Those MPs include Brexiters such as Dennis Skinner, John Mann, and the Birkenhead MP Frank Field, who resigned the Labour whip in August. They also include MPs in strong leave-leaning seats, such as Caroline Flint, Gareth Snell and Laura Smith.

Snell, the MP for Stoke Central, told the Guardian he had “huge reservations about the practicalities and politics of holding a second referendum and that is a view that is shared by more and more Labour MPs”.

The MP, who defeated the then Ukip leader, Paul Nuttall, in a byelection in 2016, said it was imperative Labour had a Brexit plan ready in the event of a general election.

“The fragility of the government along with their obvious lack of majority means we are more than likely heading toward a general election,” he said. “But that leaves a huge hole in Labour’s plan about what we would actually propose to the country regarding Brexit in that vote.”

Several Labour sources have told the Guardian that discomfort with a second referendum extends beyond those MPs who have publicly opposed both a referendum and softer Brexit options, such as membership of the European Economic Area (EEA).

One MP who was previously an enthusiastic remainer said: “I feel distinctly uncomfortable about it, having stood on a manifesto saying we would deliver Brexit to my constituents who voted leave.”

The MP estimated 20 to 30 MPs would refuse to vote for a second referendum, even if they were whipped by the Labour frontbench to do so.

Another Labour source said several people in the leadership team, including the shadow Brexit secretary, Keir Starmer, remained scarred from the debate around the EEA amendment to the EU withdrawal bill, which saw the party split three ways when MPs were whipped to abstain.

The vote against the amendment by 15 Labour rebels had led the team to believe there was privately more opposition to the idea of a second referendum than is publicly acknowledged.

“One of the key reasons for being cautious is that it’s by no means certain that there is a majority for this,” the source said.