Pro-Brexit UK lawmakers said they would read the fine print and wait for the judgment of Britain's attorney general before deciding how to vote on Tuesday.
Mr Cox wrote in a three-page conclusion that the risk of Britain ending up trapped in the bloc's trade arrangements had been reduced by May's last-minute deal but still remained.
The historic bruising for May had led her to seek changes to the agreement in order to table a second "meaningful vote" before Parliament that would stand a better chance of being voted through.
With EU leaders warning there would be no more changes or negotiations, and less than three weeks to go until the United Kingdom is due to leave, British lawmakers were facing a stark choice: support a deal many consider inadequate or run the risk that Brexit might happen chaotically, or not at all.
Many Brexiteers anxious that the backstop, aimed at avoiding controls on the border between the British province of Northern Ireland and EU-member Ireland, could trap the United Kingdom in the EU's orbit indefinitely.
Speaking to journalists shortly before the legal advice was released, Tory MP Mark Francois, the deputy chairman of the European Research Group of Brexiteers, said Mr Cox's legal advice would "have to be pretty amazing" to convince him to back the deal.
And senior Conservative Leaver John Whittingdale told the Commons Brexit Committee that the Attorney General's advice was "pretty terminal" for Mrs May's plan.
If the package passes the Commons, leaders of the 27 remaining EU states will be asked to endorse the new documents at a scheduled European Council summit in Brussels on March 21-22, before the final step of ratification by the European Parliament.
"Today is the day", Downing Street quoted May as telling her Cabinet.
The U.K. and the European Union have agreed to "legally binding changes" to the European Union withdrawal agreement and the political declaration, British Prime Minister Theresa May said Monday night. "Because there will be no third chance", Juncker warned.
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May might even try a third time to get parliamentary support in the hope that hardline eurosceptic MPs in her Conservative Party, the most vocal critics of her withdrawal treaty, might change their minds if it becomes more likely that Britain might stay in the European Union after all.
The Opposition Labour Party, meanwhile, has declared that the British PM has secured nothing new.
Another defeat on Tuesday would tee up additional votes on the way ahead.
Pro-Brexit politicians in May's Tory party insist that the plan - known as the backstop - threatens to trap the United Kingdom inside the EU's trade regime forever, because it would be impossible for Britain to leave.
That option is fraught with economic dangers and is backed only by hardcore proponents of the divorce.
Other EU nations had urged British politicians to seize the chance to back the deal and ensure an orderly departure.
Any postponement may have to be short-lived.
MPs are due to vote on the prime minister's deal from 7pm.
The PM said: "What we have secured is very clearly that the backstop can not be indefinite".
If the deal is voted down again, on the following two days lawmakers are expected to discuss and vote on two amendments.
"A positive vote tonight can remove that cloud and restore confidence and optimism in Britain, Ireland and across the European Union".