Sir Keir said that if it was voted down, Labour would table a no confidence motion in an attempt to force a general election.
The Government is to publish a command paper on the legal advice on the Brexit deal, the Prime Minister's official spokesman said.
According to the Sunday Times, Cox told ministers in a letter that the only way for the United Kingdom to withdraw from the Northern Ireland "backstop" arrangement under May's proposed deal would be to sign a new trade deal, a process that would likely take years to complete.
Tuesday was to have seen the start of the debate proper on the exit deal, which is due to culminate in a vote early next week.
'I don't think there is any chance of pulling the vote.
Senior MPs from six parties wrote to Mr Bercow asking him to begin contempt proceedings against the Government after full publication of the attorney general's findings was withheld "in the public interest".
It has threatened to initiate proceedings against the government for contempt of parliament, a move that could potentially result in one or more ministers being suspended or expelled from the House of Commons, the lower chamber.
With the odds looking stacked against her, May is touring the country and media studios to try to win over critics including both eurosceptics and europhiles who say the deal will leave Britain a diminished state, still linked economically to the European Union but no longer having any say over the rules.
Lawmakers can send the issue to a committee with the power to sanction ministers.
"And it seems to me a situation where we're now reliant on our learned friends to take cases in worldwide court, rather than this sovereign parliament being able to decide when we can get out of these arrangements".
The Prime Minister insisted she would still have a job after the crucial Commons vote on December 11.
Speaking in Buenos Aires on Saturday, May said: "The next nine days are a really important time for our country leading up to the vote on this deal".
"This is the deal that delivers for the British people", May will tell MPs.
He added: "Even at this eleventh hour, I would urge ministers to step back from the brink and to not go down in history as the first Government to be found in contempt".
Ministers have been accused of ignoring the will of the House after saying they will publish only a "full reasoned political statement" on the legal position concerning the Withdrawal Agreement.
MPs from across Parliament have argued that they are required to release the full advice - including Mr Cox's legal opinion - following a binding vote last month.
In particular, he conceded there were difficulties with the "Irish backstop" section, created to prevent a border between Ireland and Northern Ireland if the as-yet-unnegotiated trade deal between Britain and the European Union doesn't solve the problem.
Under parliamentary rules, it is up to the Speaker, who chairs debates in the House of Commons, to decide whether to allow a contempt motion to be voted upon.
"I make no bones about it, I would have preferred to have seen a unilateral right of termination in this backstop", he said.
"But I'm prepared to lend my support to this agreement because I do not believe that we're likely to be entrapped in it permanently".
In his letter, Mr Robbins advised that extending the transition period would give a "more cast iron escape" from European Union customs rules than the backstop.