Jeremy Corbyn is today coming under fresh pressure to back a softer Brexit as a new poll shows an overwhelming majority of members of the Unite super-union – Labour’s biggest financial backer – believe the current Labour position in favour of leaving the single market will be bad for jobs.

The survey by YouGov, released before Unite’s annual conference this week, also shows that most of the union’s members back giving the people a vote on any deal Theresa May brings back from Brussels – something Labour also opposes.

The survey is released as sources within grassroots movement Momentum say senior figures at the top of the organisation are now keen to ensure a full debate on Brexit at the Labour conference in September, opening a possible way for anti- and soft-Brexit members and delegates to force a change to official policy.

The pressure on Corbyn and shadow Brexit secretary Kier Starmer has also been mounting within parliament. Earlier this month, 89 Labour MPs went against instructions and voted for an amendment calling on the government to back membership of the European Economic Area, which would allow participation in the single market.

The YouGov poll, commissioned by the People’s Vote Campaign, found that Unite members believed by a margin of more than three to one (57% to 18%) that leaving the single market would be bad for jobs. When asked if they believed Britain would be worse off outside the single market, 58% said it would, while 21% took the opposite view. Some 57% of the 913 Unite members questioned backed a people’s vote on the final outcome against 34% who did not, and 24% of those who voted Leave backed giving the people another say.

Corbyn maintains that he is determined to pursue a “jobs-first Brexit”. But the polling suggests many in the union movement now think his policy will have the opposite effect. Labour is proposing a new relationship with the single market that it says would give the UK access but fall short of full membership. The model has been dismissed by Brussels as unrealistic as it tries to keep all the advantages without having to comply with the EU’s four freedoms, including free movement of workers.

Many of the motions submitted to the Unite conference depart from Labour’s policy and express support for the single market, and some for a people’s vote. One motion, submitted by West Midlands car workers to the conference, states: “Only now have voters started to see through the initial referendum campaigns and truly understand the implications and costs of leaving the EU. The final negotiations of UK Brexit terms cannot be left for parliament alone to agree upon because the facts presented to the voting public have changed so much since the referendum vote. The British electorate must ultimately have the final say.”

Last week, another group, Labour for a People’s Vote, which is led by several former Momentum figures as well as trade union leaders and has the backing of more than 60 constituency Labour parties, announced that it would try to force a vote at this year’s party conference to change Corbyn’s Brexit policy.

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The group will call for Labour to oppose the government’s final Brexit deal. The party has said it would do this if the deal with the EU did not meet six tests set by shadow Brexit secretary Starmer. The group will then campaign for the party to go further and campaign for a public vote on the deal with an option to stay in the EU should voters reject the deal.

Baroness Margaret Prosser, a former deputy general secretary of the Transport and General Workers Union, which merged into Unite in 2007, said: “It’s becoming increasingly clear that the Tory government is hell-bent on a bad Brexit deal for British workers that will damage jobs, public services and the future of young people. We are already seeing worrying signs about the impact on Airbus, carmakers and other key sectors of the economy. In these circumstances, workers look to our union to stand up for their interests.”

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Len McCluskey, Unite’s general secretary, is a close ally of Corbyn, but only 3% of his members said in the survey that the union should base its policy on what’s best for the Labour party, with 84% saying it should be driven instead by what’s best for either the country or for members of Unite.