The PM says the advice is confidential. but some MPs think ministers do not want to admit it says the UK could be indefinitely tied to EU customs rules.
Ex-foreign secretary Boris Johnson has joined calls for its publication, which critics say could sink the PM’s deal.
Attorney General Geoffrey Cox will make a statement about it on Monday.
He is set to publish a reduced version of the legal advice – despite calls from MPs from all parties to publish a full version.
His statement to the House of Commons will be followed by five days of debate on the deal.
MPs say the statement from the attorney general does not respect a binding Commons vote last month, which required the government to lay before Parliament “any legal advice in full”.
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Labour is planning to join forces with other parties, including the DUP, who keep Mrs May in power, to initiate contempt of Parliament proceedings unless the government backs down.
Shadow Brexit secretary Sir Keir Starmer told Sky News: “If they don’t produce [the advice] tomorrow (Monday) then we will start contempt proceedings. This will be a collision course between the government and Parliament.”
His shadow cabinet colleague Barry Gardiner told the BBC’s Andrew Marr show the prime minister faced a “very serious constitutional crisis” if she refused, and the only answer was a general election.
Brexit-supporting MPs say it could mean an open-ended commitment for the UK, forcing it to remain in the EU’s customs union while details of the deal are being worked out.
Downing Street has not responded to claims by the Daily Telegraph that a letter from Mrs May’s chief Brexit adviser, Olly Robbins, warned her it was a “bad outcome” with no legal “guarantee” Britain would be able to exit the mechanism.
Mr Johnson, who resigned from government over the PM’s Brexit vision, described the arrangement in his weekly Daily Telegraph column as “a great steel trap that is about to clamp its jaws around our hind limbs and prevent our escape”.
The DUP’s Sammy Wilson said it was important for MPs to know exactly what they are voting for and the implications for Northern Ireland.
“What we need to know is what legal advice has been given to the government about the extent of how Northern Ireland will be stuck in the backstop… [and] how do you get out of this backstop,” he said.
The Commons debate on Theresa May’s deal is due to get under way on Tuesday.
But if contempt proceedings were begun by Labour, it would be up to Commons Speaker John Bercow to decide whether a debate and vote should be held first.
‘Fight of her life’
Mrs May faces the fight of her political life to get MPs to back the deal in a Commons vote on 11 December.
Ministers insist they can persuade enough of them to change their mind because, they claim, the alternative is a no-deal Brexit or “no Brexit at all”.
On Sunday, the Brexit-supporting Environment Secretary Michael Gove warned MPs there could be another EU referendum if they failed to back the PM’s deal.
And on Monday, Home Secretary Sajid Javid will attempt to sell the deal by stressing that it will bring an end to EU free movement in the UK and replace it with a skills-based immigration regime.
“In just over a week’s time, MPs will vote on the Brexit deal the government has secured. My message to them is this: let’s take this chance to deliver an immigration system truly underpinned by public support,” he will say.
Meanwhile, support for a further referendum is growing – on Saturday science minister Sam Gyimah quit the government to join the People’s Vote campaign.
And a cross-party group of 17 MPs, in a letter published in the Observer, also called for Parliament to support another referendum at the earliest opportunity.