And the move is threatening to take the gloss off the deal agreed by ministers at Chequers last week, whereby Britain would attempt to negotiate a process of “managed divergence” from EU rules.
The motion – put down by Tory MP Nicky Morgan, who has joined forces with Labour’s Yvette Cooper – would voice support for remaining in the customs union.
The Commons Liaison Committee, which consists of all 35 heads of Parliamentary Select Committees, is trying to arrange a vote as soon as possible.
The motion is being proposed after the Government delayed a vote on its Trade Bill until May after an amendment was put down which would have bound her to stay in the customs union.
Labour MP Chuka Ummunna, who is working alongside Tories Anna Soubry and Jonathan Djanogly to push for a vote, said: “It is undemocratic for the Prime Minister to put off a vote.
“She has no majority for taking us out of the EU customs union but continues to negotiate as if she has one.
“If she delays these important votes, we will seek to instigate a vote from the backbenches to illustrate Parliament’s position on this issue.”
Mrs May had attempted to strike an optimistic note after the agreement thrashed out at Chequers on Thursday, saying the Brexit deal “must present an ambitious future for our great country”.
She added: “The decisions we make now will shape this country for a generation.
“If we get them right, Brexit will be the beginning of a bright new chapter in our national story, and our best days really do lie ahead of us.”
But ministers are said to be split on the issue, with foreign secretary Boris Johnson apparently hailing it as a “victory”, and Michael Gove, Secretary of for Environment, Rural Affairs and Food, doubtful the EU will back the proposal of “managed divergence”.
Chancellor Philip Hammond is also said to be sceptical.
Guy Verhofstadt, the EU’s Brexit negotiator, seemed to confirm this stance yesterday when he said: “The plan appears to be one to keep the Tory party together rather than a realistic negotiating position.
“Selectively, picking which EU rules to follow, while seeking to lower standards in other areas, will be rejected.”