British lawmakers who favour leaving the European Union say the Prime Minister's deal keeps Britain bound too closely to the bloc while those who want to maintain close ties argue that it creates barriers between the United Kingdom and its biggest trading partner.
The day before the vote, on December 10, the European Court of Justice of Justice will deliver a judgment on whether Britain can unilaterally reverse its move to leave.
MPs backed the amendment - put forward by Dominic Grieve - by 321 votes to 299 as they headed into five days of debates on the Brexit withdrawal agreement ahead of the crucial vote on Tuesday next week.
Under the current deal the "backstop" would come into effect at the end of the Brexit transition period, which is due to last until the end of 2020 but may be extended.
He said he had already been given authority by his local party's executive to submit a letter to Sir Graham Brady, the chairman of the party's backbench 1922 Committee, seeking a vote of no confidence in Mrs May.
But many members of Parliament worry that the transition period is a bridge to nowhere.
On Monday, the government's chief legal adviser, Attorney General Geoffrey Cox, had published a summary of his legal advice, but that did not satisfy MPs who said the advice should have been published in full.
"For these reasons I will not be supporting Mrs May's deal, and I believe if it is defeated, that Parliament need to consider other options, including potentially a General Election, further renegotiation either by this or another Government, and the option for a possible referendum on the deal".
The analysis by Attorney General Geoffrey Cox warns that under the terms of the Irish backstop arrangement, the United Kingdom will be trapped in a series of "protracted and repeated rounds of negotiations" in the years after the United Kingdom leaves the EU.
The Brief – Brexit purgatory waiting room blues “Would you please consider voting for this deal so we can all get on with our lives.”
"All week we have been told by government ministers that releasing this information could harm the national interest, but the advice contains nothing of the sort, ' he added".
Before the prime minister appeared at the despatch box, her government had gone down to defeat for the third time in an hour.
The backstop is intended as a temporary measure, but pro-Brexit lawmakers say it could leave Britain tied to the European Union indefinitely and unable to strike new trade deals around the world.
"This risk must be weighed against the political and economic imperative of both sides to reach an agreement that constitutes a politically stable and permanent basis for their future relationship".
Concerns about the backstop are a key driver of opposition to the deal among both May's own Conservative lawmakers and the Northern Irish Democratic Unionist Party (DUP), which props up her minority government.
A Downing Street spokeswoman rejected suggestions that the Withdrawal Agreement could be tweaked ahead of next week's vote to take account of concerns expressed by MPs.
She told BBC's radio program that she was talking to her colleagues about their concerns over the Northern Irish backstop, on which experts say the December 11 parliament vote on the Brexit deal would hinge.
The UK government became the first in the nation's history to lose a contempt of Parliament vote over its handling of legal advice about its proposed Brexit deal.
May faced the prospect of "hitting a brick wall at speed" and predicted that no consensus would found in the House of Commons on Brexit.
International Trade Secretary Liam Fox said Wednesday there was "a real danger that the House of Commons, which has a natural "Remain" majority, may attempt to steal Brexit from the British people".