Ahead of the crucial votes on the EU Withdrawal Bill, Brexit Secretary David Davis warned MPs that defeat would undermine the UK's negotiating stance in Brussels. MPs will vote again on the issue later in the summer.
A series of further votes will take place on Wednesday, with no defeats expected after ministers agreed compromise wording over post-Brexit plans for a "customs arrangement".
Party officials have been frantically lobbying lawmakers to try to persuade those who have threatened to vote against the government to stay in line, using arguments ranging from the threat being turfed out by a government led by opposition Labour leader Jeremy Corbyn to losing any leverage in Brexit talks.
May warned her MPs: "The message we send to the country through our votes this week is important".
The disagreement centers on whether the government agreed to consider a specific clause of the rebel proposal that would hand parliament control of the Brexit process if ministers are unable to strike an exit deal by February 15, 2019.
More than a dozen MPs who were called into the prime minister's office in the final minutes of the Brexit debate on Tuesday believed they had received assurances the government would discuss clause C of an amendment tabled by the former attorney general Dominic Grieve.
"It would tie the hands of the government in a way that I think could make no deal more likely".
It would reduce the risk of the United Kingdom exiting the European Union without a deal, as it means that MPs could insist that the government go back to negotiating table.
The front pages of Leave-backing British newspapers said accepting the amendments would betray the 52 percent who backed Brexit in the seismic 2016 referendum. The government says leaving the customs union will free the country to strike trade deals around the world.
May is struggling to unite not only her party but her top team of ministers over how to leave the European Union, particularly over the future customs arrangements which have pitted those wanting closer ties with the European Union against others who demand a clean break.
This amendment had looked unsafe for the prime minister.
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Magnusson joined City from Juventus after Euro 2016 and scored his only goal for the Robins on his debut in August that year. The win 1-0 win over Morocco was a highlight for the national team and takes pressure off the team's chances of advancing.
But, the pro-EU MPs' version of what they were promised appears to differ from what they government says it offered, threatening to reignite the dispute and reviving the possibility of a revolt that would badly damage May's authority.
Pro-EU Conservative Heidi Allen said it was inevitable "we will have to come to a customs union agreement", even if it was given another label.
Another Conservative Remainer, former Cabinet minister Nicky Morgan, denied that the rebels had been "played" by the Prime Minister.
The Bill will return to the Lords on June 18, kicking off a process of "Parliamentary ping-pong" which sees it bounce from House to House until agreement is reached.
A total of 89 of the party's 257 MPs ignored front bench orders to abstain from voting either for or against an amendment to the EU (Withdrawal Bill) that sought to prioritise European Economic Area (EEA) membership in EU negotiations.
Labour opposed the EEA model and introduced its own amendment, tabled by Sir Keir Starmer, which was defeated by 82 votes on Wednesday night.
Theresa May's Janus-faced promise on a "meaningful vote" may have averted a defeat.
However, the party is divided on the issue, with a total of 89 Labour MPs choosing to defy Corbyn.
She will also be tested on Wednesday by rebels in her own party over her commitment to leave the EU's single market and customs union, which will transform Britain's future trading relationships for many years to come.
Caroline Flint, Labour MP for, argued against both EEA proposals, saying "there has to be an end to freedom of movement" when Brexit comes into force.
"I am trying to negotiate the best deal for Britain".