Or EU leaders could throw a lifeline to May by offering a deal on the backstop.
The pound extended earlier gains as Mrs May won legally binding Brexit assurances from the European Union, in a last ditch attempt to sway rebellious MPs who have threatened to vote down her divorce deal in a parliamentary vote on Tuesday.
Effective July 2018, access to full reports will only be available with a subscription. The "Irish backstop" is an insurance policy that will ensure there is no hard border in Ireland.
So she also needed to look to the opposition benches. As said by Mrs May last week, rejecting her deal could lead anywhere - including not leaving the European Union at all.
The government was even weighing up whether to scrap Tuesday's planned vote on the basis that no significant changes had been secured.
Intensive discussions led by the Labour MPs Stephen Kinnock and Lucy Powell are under way over the wording of a potential backbench amendment that might be close enough to Labour's version of Brexit to allow the leadership to support it.
After Parliament rejected the deal in a defeat of historic proportions in January, May promised to seek changes to address these concerns.
Meanwhile, JR Zhou, market analyst at Infinox, said markets were "convinced" that the deal does not have the numbers to pass and added that May's speech to the Commons "has now been thrust firmly into Hail Mary territory - a last-ditch effort to sell a seemingly unsellable deal to MPs". "These are things any responsible politician should care about", Mr Juncker said.
"The legal risk remains unchanged that if through no such demonstrable failure of either party, but simply because of intractable differences, that situation does arise, the United Kingdom would have, at least while the fundamental circumstances remained the same, no internationally lawful means of exiting the protocol's arrangements, save by agreement", he said. "We have a deal on the table which does exactly do this". "Tomorrow, the House of Commons will debate the improved deal that these legal changes have created".
The changes fall short of demands of hardcore UK Brexiteers, but may persuade some politicians to switch their votes.
Following technical talks over the weekend, both sides were expected to go over the phrasing in the part of the agreement dealing with the post-Brexit border on the island of Ireland.
Most observers see it as highly unlikely the deal will pass through parliament and Mrs May's spokesperson has been forced to deny rumours the prime minister may delay the vote with only a couple of weeks to go until the United Kingdom leaves the EU.