That's because May has been unable to persuade Brussels to change the agreement, which her team negotiated past year, and put in legally binding reassurances that the United Kingdom will not become trapped in a long-term customs arrangement with the European Union.
European Commission President Jean-Claude Juncker tweeted that the changes provide "meaningful clarifications and legal guarantees".
May's deal was struck during more than a year of tough negotiations, and covers Britain's financial settlement, expatriate rights, the Irish border and plans for a transition period. After May called a snap election in 2017 and led her party to disastrous results, George Osborne, the former chancellor of the exchequer whom May had sacked, called her a "dead woman walking."
"The British Prime Minister is traveling to Strasbourg this evening I understand to try to finalize an agreement if that's possible", Coveney told reporters in Dublin.
Nigel Farage, who spearheaded the campaign that led to Britons voting in the June 2016 referendum to leave the European Union, held firm.
But if they reject it, MPs could vote on Thursday to delay Brexit - giving further time for pro-Europeans to try to undermine the whole process. Prime Minister Theresa May's government has been seeking changes, but the European Union refuses to reopen the agreement after long negotiations.
Meanwhile, the Opposition Labour Party has been mounting pressure on the government for an official update to Parliament on the talks with the EU.
Ministers concede the deal is not flawless but say it is the best way to move forward, and warn rejecting it could put Brexit at risk.
Due to the sensitive nature of the region and because the free passage is one of the crucial articles of the 1998 Belfast Agreement, the U.K., EU, Northern Ireland and Ireland all reject the idea of returning to a hard border where checkpoints and customs buildings will need to be installed.
The Climb in GBP/USD Stops at the 200 SMA
Mr Lidington said the Attorney General would be publishing advice ahead of Tuesday's debate, but needed time to consider the new changes.
The airline is to put plans in place to protect themselves from a no-deal and wants to ensure they will remain a majority European Union controlled and owned airline.
For pro-Brexit campaigners, that would be a betrayal of the 2016 referendum vote to leave the EU.
May has said if she loses the vote on Tuesday, there will be further votes on Wednesday on whether the United Kingdom should leave with no-deal and on Thursday on whether they should seek an extension to the Article 50 withdrawal process.
The third, another referendum on Brexit, has lost momentum in recent weeks, but that campaign could be reenergized if Brexit is postponed this week - not least because the minister in charge of Brexit has held talks with lawmakers in favor of a second referendum in the past few days.
Without any changes to the deal, she could face another heavy defeat.
Media captionBig Brexit moment: Will MPs back or bin the PM's deal?
The exact details of this vote are not yet known, including which way May will instruct her Conservative Party to vote. "Tuesday is about exposing the weakness of the prime minister".