The People’s Vote campaign has issued a warning to Labour MPs tempted to vote for any Brexit deal struck by Boris Johnson that they would be held responsible for a rightwing assault on Britain’s public services and economy.

The campaign circulated a dossier warning that a Johnson deal would lead to a “bonfire of rights”, as it emerged that two Labour MPs, Stephen Kinnock and Caroline Flint, had led a delegation to meet EU negotiator Michael Barnier on Thursday to discuss their potential support for a withdrawal deal.

The warnings were led by Margaret Beckett, the co-chair of People’s Vote and former foreign secretary, who said MPs voting for a Johnson deal would be facilitating the “stuff of rightwing Tory dreams”, including deregulation and dependence on Donald Trump’s America.

There are increasing worries among campaigners for a second referendum that a bloc of Labour MPs with leave-voting seats may be prepared to back a Brexit deal if Johnson manages to secure one in return for some assurances on workers’ rights and environmental protections.

The new MPs for a Deal group, led by Kinnock and Flint, believes dozens of opposition MPs could be persuaded to back a version of the withdrawal agreement bill combined with some of the promises that May had offered the Labour leadership during her unsuccessful talks before she was ousted in the summer.

To get legislation through parliament the prime minister will almost certainly need more Labour support unless he gets almost every Eurosceptic Conservative MP to vote with him, including the most hardline members of the European Research Group, who want to leave with no deal.

With Johnson still insisting he wants a deal, the new dossier produced by the People’s Vote campaign was circulated to Labour MPs on Thursday night. It argued that any agreement struck by the current government would lead to a bonfire of rights, a race to the bottom on standards, backwards steps on climate change and the NHS put up for sale. The document is designed to warn Labour MPs that a no-deal Brexit is not the only danger.

Beckett said: “No one should trust Boris Johnson when he says he wants a Brexit deal. But nor should anyone trust him if he does eventually come back with a some sort of deal.

“Indeed, any MP who might be tempted to back it needs to recognise the likelihood that a Johnson Brexit will destroy jobs, undermine public services and usher in the kind of offshore deregulated pirate economy which is the stuff of rightwing Tory dreams – and our nightmares. That is certainly the aim of his supporters who would love to claim Labour support – and blame us afterwards for the consequences.”

Tom Watson, the Labour deputy leader and a supporter of the People’s Vote campaign, added: “I know some Labour MPs are saying they want to vote for a deal to end this Brexit crisis. But a Boris Johnson Brexit deal would be the culmination of a decade-long project which is of the right, by the right and for the right. Any deal he negotiates will mean the deregulation of our economy, a race to the bottom in standards, less money for our public services and our NHS put up for sale to US companies.

“I hope Labour MPs will stay united in defence of our values and back a people’s vote on any Brexit deal, which is the only democratic way to solve this crisis.”

Flint and Kinnock told reporters they had come to Brussels on Barnier’s request where the EU’s chief negotiator had asked them about the number of Labour MPs who would now back a deal.

Flint said: “He asked us about the group. He asked us about the state of play in the UK … There’s a real sense, there’s a number of parliamentarians, cross-party, trying to get a deal before 31 October and that’s why this group has been formed.

“And, in terms of that, what different pressure we can put on … to get parties to really work towards that deal … And that was part of what he was interested in.”

The two Labour MPs said they believed there were potentially “20 to 30” of their party colleagues who would back a deal similar to that tabled by Theresa May, with additional safeguards on workers’ rights and environmental standards.

Kinnock said: “I think I speak for a lot of Labour colleagues as well who don’t have big issues with the backstop at all. The reason we predominantly voted against the deal – Caroline did vote for it – was around concerns of the political declaration on the future relationship. I think we have done the cross-party talks and from them I think emerged a package of excellent concessions to our position that I think we should have jumped at and I think it was a tremendous mistake to have walked away from those talks.”

He added: “On numbers, 92% of the current Conservative parliamentary party have voted for a deal at one point or another and 92% have voted for the backstop. It’s a question then of how many hardline Brexiteers are prepared to rebel against a deal that has a three-line whip attached to it. This is a coalition for a deal. Different parts of the coalition are looking for different things but we really are now at the last roll of the dice.”