Securing a vote on the trading relationship would mean MPs can effectively veto terms that threaten industry and see the UK market flooded with sub-standard food, goods and services.
Any moves by May to push through her deal with Labour votes is likely to infuriate Tory MPs from the hardline eurosceptic ERG (European Research Group), who will view it as an attempt to frustrate a ‘clean’ Brexit.
After May emerged victorious from a no-confidence challenge before Christmas, however, Tory party rules mean MPs cannot launch another attempt to topple her until December, and Labour backbenchers are hopeful they can wrest more control over Brexit.
Labour MPs are split over the issue, however.
Pro-Brexit Labour MP John Mann, who was invited for talks at Downing Street last week, said civil servants raised the prospect of a multibillion investment fund for Brexit areas with him – something which his fellow MP Anna Turley said amounted to May “bribing” Labour.
In what appeared to be a thinly-veiled swipe at would-be rebels, Shadow Trade Secretary Barry Gardiner said in an interview on Sunday his “vote is not for sale”.
After trade union bosses, such as Unite’s Len McCluskey, entered Downing Street last week for talks, other Leave-minded MPs believe potential rebels should hold out for more.
Labour’s Gareth Snell, who represents the pro-Brexit Stoke Central, was among the MPs who voted against moves by Yvette Cooper to delay Brexit last week,
He predicted more MPs were preparing to defy Corbyn and back May’s Brexit deal before exit day on March 29.
“I think the numbers are growing but I could not give you an exact number on how many are where I am,” he said.
“I think the number will continue to grow the nearer we get to exit day.”
He added: “We all have to coalesce around something and that includes the hardcore second-referendum MPs.
“The pragmatic middle will find a deal that mitigates the economic damage, leaves EU institutions, delivers on the spirit of the referendum and still allows us to be a trading nation.”
HuffPost UK has contacted Number 10 for comment.