If the deal is voted down, Britain risks crashing out of the European Union on March 29 with no agreement in place, a messy outcome that could plunge the country into its worst recession for decades.
It is not certain, however, that MPs will get the chance to vote for the change.
A deal setting out the terms of the United Kingdom's divorce from the European Union has been agreed to between the prime minister and the European Union, but it must pass a vote in Britain's Parliament before it is formally adopted.
Following the cancelled vote, May survived a leadership challenge triggered by disgruntled Eurosceptics in her party and has been back to Brussels to try and get additional reassurances about the so-called Irish backstop which was a major sticking point in opposition to the deal.
Critics of the backstop, which proposes that the whole of the United Kingdom, including Northern Ireland, remain in a customs union with the EU "unless and until" the bloc agrees there is no prospect of a return to a hard border, argue it could tie the United Kingdom into the EU's orbit indefinitely.
The EU, for its part, has insisted that the withdrawal agreement can not be renegotiated but has made clear the backstop is meant only as a temporary measure of last resort.
The DUP's deputy leader Nigel Dodds said the Withdrawal Agreement remained "toxic" as a result of the Irish backstop measure.
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"The coming days will show if this government is made of the right stuff", he added.
Analysts said May's comments to the BBC did little to hide the fact the British leader has no "gamechanger" amendments to the proposed withdrawal agreement capable of swinging the parliamentary arithmetic in her favour.
"I think some people within the People's Vote movement seem to think that their goal is to slap the Labour Party around", the shadow foreign secretary said.
"I don't think anybody can say exactly what will happen in terms of the reaction we'll see in parliament".
But she admitted she had no way of knowing what might happen if MPs rejected her deal, adding that the country would be in uncharted territory.
"The only way you're going to get on and deliver Brexit is what's called a "no deal" Brexit".
In a letter published by United Kingdom newspaper the Mail on Sunday, May warned critics of her departure plan risk damaging Britain's democracy and weakening its economy by opposing her deal.
Pressure has intensified after a large-scale opinion poll commissioned by the People's Vote campaign indicated Labour could suffer a backlash from voters if it failed to oppose Brexit in Parliament.
The survey by polling firm YouGov showed that if a referendum were held immediately, 46 per cent would vote to remain, 39 per cent would vote to leave, and the rest either did not know, would not vote, or refused to answer the question.