On Wednesday, Parliament voted for an amendment calling on the government to set out its Plan B within three working parliamentary days of the January 15 vote, rather than the 21 days specified in Brexit law.
In a sign of the anger and division Brexit has sown among politicians, the motion prompted bad-tempered scenes in Parliament, as Conservatives accused Speaker John Bercow of contravening parliamentary convention by allowing a vote on the amendment.
"We are doing everything we can to win the meaningful vote that happens on Tuesday", the spokesman said.
The No 10 source said the Prime Minister had always meant to "respond quickly" if she fails to secure the support of the Commons. The scale of the defeat suffered by the deal in the first vote is key: if the scale of the defeat is marginal then markets could bid Sterling higher on the assumption it will cross the line in the event of some fresh concessions being offered by the EU.
May told reporters that the only way to avoid such a scenario was for lawmakers to support her deal.
The package, tabled by former minister Sir Hugo Swire, also places a legally-binding commitment on the Government to end the backstop arrangement within 12 months, and to seek assurances from the European Union that it will seek to do the same.
The "backstop" is a fallback option to avoid a hard border between the two if a trade deal is not agreed at the end of the transition period (December 31, 2020).
It could also open the door to alternatives, such as a referendum.
"I am committed to getting the best outcome for this country, which is supporting the prime minister's deal".
The campaign group says that with three-quarters of Labour voters in Yorkshire wanting the United Kingdom to stay in the European Union, its vote share is slipping as Remainers turn their backs on the party.
While 308 lawmakers supported the amendment, 297 voted against it.
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Japan's ambassador to Britain, Koji Tsuruoka, issued a blunt warning about Brexit last February when he said Japanese companies would have to leave Britain if trade barriers made them unprofitable.
A no-deal Brexit remains the default option if the Parliament doesn't back May's agreement on January 15, even though a majority of lawmakers supported a motion on Tuesday created to reduce the chances of Britain tumbling out of the bloc on 29 March.
Several MPs challenged the Speaker with points of order, with Tory Brexiteer Peter Bone asking for an explanation as to why, when he had attempted to propose an amendment to the same motion, he had been refused by the Table Office, saying "I was told it would be totally out of order and there would be no other amendments filed".
Ministers argued that the Government-tabled motion was not amendable and so Mr Grieve's amendment should not have been tabled for a vote.
Mr Clark had earlier said that after the Government suffered two Commons defeats over its Brexit plans in the space of 24 hours, it was clear that there was no majority for leaving the European Union without an agreement.
It highlighted that if parliament does not agree with Mrs May then it must immediately outline its plan to avert a no deal and secure British jobs.
Justice minister Rory Stewart questioned the basis of Mr Bercow's decision, telling the BBC: "It is a very, very unusual thing that he did". I genuinely think we can't do it on March 29 this year.
So even if he persuades a few wobbly Brexiters to vote with the PM on Tuesday, he has guaranteed that Tory MPs who want a referendum will not be wooed back to her cause.
"I just invite you to reflect on the conclusion that many of us will have inevitably have come to".
Speaking to MPs earlier, May said: "The only way to avoid no deal is to vote for the deal".