In a two-way choice, 64% of grassroots members would opt for a “no-deal” Brexit on 29 March, with just 29% preferring the plan Theresa May has negotiated.

In a three-way choice, 57% said they would support leaving under the potentially chaotic no-deal scenario.

Under a quarter – 23% – would back Theresa May’s deal, and 15% said they would prefer to remain in the EU. Five percent had no preference.

The YouGov poll, funded by the Economic and Social Research Council, comes as MPs prepare for a crunch vote on Mrs May’s plan – due the week starting 14 January.

The vote was due to be held before Christmas but was pushed back at the last minute when it became clear the plan would easily be defeated.

Mrs May’s attempt to get more guarantees over the controversial Irish border “backstop” – the issue troubling many MPs – was dismissed by European leaders.

The backstop may never be needed, but is intended to prevent a hard border in Ireland by ensuring the UK abides by EU customs rules if a future trade deal cannot be agreed.

The government is poised to launch a publicity drive to prepare the public for the possibility of no-deal as the weeks tick down to Brexit Day.

:: Withdrawal deal in detail: What you need to know

There have been other setbacks in recent days for Mrs May, with DUP deputy leader Nigel Dodds saying his party’s concerns over the backstop were not eased after a meeting with her on Thursday.

Mr Dodds said: “The Withdrawal Agreement, as currently proposed, flies in the face of the government’s commitments on Northern Ireland as we leave the EU.”

Ireland’s leader Leo Varadkar has also insisted there can be no changes to the backstop and that he and German chancellor Angela Merkel had agreed to stick to what was previously agreed.

He said: “We’re happy to offer reassurances and guarantees to the UK, but not reassurances and guarantees that contradict or change what was agreed back in November.”

The European Commission also appeared to shut down Mrs May’s chance for any last-minute tinkering with the deal, saying “no further meetings are foreseen” with the UK because negotiations have ended.

Meanwhile, a warning over the potential fallout from a no-deal Brexit has been issued by groups representing 150 universities and colleges.

They say it could cause “an academic, cultural and scientific setback from which it would take decades to recover”, affecting scientific research into areas such as cancer treatment and climate change.

Professor Dame Janet Beer, head of Universities UK, warned that “world-leading academics and researchers” may leave for countries with access to EU funding programmes – or avoid coming altogether.

She said: “We are home to one of the best research systems in the world, attractive to stellar academics, top students and global partnerships, and we must not let this be compromised by a no-deal Brexit.

“Time is running out to make decisions on issues which will ultimately affect the country and society as a whole.”

The letter, signed by the heads of Universities UK, the Russell Group, Guild HE, Million Plus and University Alliance, warns 50,000 staff and 130,000 students from EU countries are facing “significant uncertainty about their futures”.

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