Thousands of Labour members have demanded their party oppose Theresa May’s Brexit deal and back a second referendum over EU membership. The call comes before a key party gathering which will be held amid warnings that some are already ending their membership over the issue.

The pressure emerges as the biggest Brexit poll conducted since the referendum suggests support for Labour would fall significantly should it back or allow its MPs to back a Brexit agreement. More than 5,000 Labour members and supporters have contacted the party before its policy meeting of senior figures this week.

Labour’s national policy forum, which includes trade union bosses, senior party officials and shadow cabinet ministers, meets on Wednesday. Officials will offer testimony that local members are quitting over the party’s refusal to oppose Brexit.

One submission, from the chair of Devizes Labour party, warns that members are resigning and threatening to resign “because of Labour’s reluctance to take a more proactive role in the campaign to force a people’s vote on this issue”. She warns: “If Labour now fails us on this there is likely to be a mass exodus of the activists we need to fight for Labour in nearby winnable constituencies.”

Activists point to the biggest ever Brexit poll, which shows that Labour’s support would collapse at the next election if it eventually backed a Brexit deal or handed its MPs freedom to vote on such a deal. The YouGov poll of 25,000 voters for the People’s Vote campaign found that Labour’s support would then slump to 26% of the vote – lower than the 27.6% secured by Michael Foot in the party’s disastrous 1983 election. Labour is currently committed to voting against May’s deal, but has stopped short of opposing Brexit under different terms.

The poll, conducted over the Christmas break, suggested there was a majority in favour of a second referendum and against Brexit. Voters would prefer that they, rather than MPs, are given the final say by 53% to 47%, excluding those who said they did not know. It found that 54% back staying in the EU, while 46% back leaving, excluding those who did not know.

A Labour spokesperson said: “As unanimously agreed at conference, if Theresa May’s botched Brexit deal is voted down in parliament then a general election should be called. In line with the policy agreed at conference, if the Conservatives block a general election then we will keep all options on the table, including the option of campaigning for a public vote.”

There is huge pressure on Jeremy Corbyn and the PM as MPs prepare to return after the Christmas break, with the crucial vote on May’s deal scheduled for a week on Tuesday. Some insiders are already expecting a further delay to the vote. Under one plan, MPs would pass an amendment suggesting the deal will only pass with further legal guarantees from the EU about the so-called “Irish backstop” – which would keep the Irish border open but could tether Britain to the EU’s customs union. It comes after a concerted push from some cabinet ministers to show Brussels what sort of compromise would be required to secure parliamentary support. Downing Street sources insisted last night that the vote on the Brexit deal would go ahead next week.

It has also emerged that Tory party members are favouring leading Brexiters as their preferred candidates to replace May. Boris Johnson, Jacob Rees-Mogg and David Davis are top of the party’s grassroots list. The next leader is likely to take the lead role in negotiating Britain’s long-term trading relationship with the EU.

The cluster of Brexiters at the top of members’ wishlist is the latest sign of pressure May faces from the right. After May pledged not to fight the next election as leader, Tory members were asked to name their preferred successor by the Party Members Project, run by Queen Mary University, London, and Sussex University.

Johnson, who has been a leading critic of May’s deal since quitting as foreign secretary, topped the poll with 20%. Rees-Mogg, the leader of the European Research Group of pro-Brexit MPs, trailed in second on 15%, while Davis, who quit as Brexit secretary last year, scored 8%. However, 12% said they did not know who should be the next leader, so the field could yet open up should May manage to hold on for some time and new figures emerge as contenders.

Worryingly for Remain supporters, home secretary Sajid Javid was the only figure who originally backed staying in the EU, among the top five names in the members’ wishlist.

Meanwhile, May has warned MPs of the risks they are taking with democracy and the livelihoods of their constituents by seeking either a second referendum or “their particular vision” of Brexit.

Writing in the Mail on Sunday, the PM was also critical of Labour’s approach under Corbyn, saying it was based on a “cynical tissue of incoherence, designed to avoid difficult decisions”.

In a message that appeared aimed at winning opposition support for her deal, she said that “MPs of every party will face the same question when the division bell rings. It is a question of profound significance for our democracy and for our constituents.

“The only way to both honour the result of the referendum and protect jobs and security is by backing the deal that is on the table.”