On the eve of Tuesday's vote, May flew to the French city of Strasbourg, where EU legislators were meeting, for nighttime talks with European Commission President Jean-Claude Juncker.
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.
Ms May said the House of Commons will debate the improved deal on Wednesday.
European officials expressed frustration with May's attempts to secure concessions with so little time left before Britain is due to leave on March 29.
The prime minister's office said a trip was not confirmed, though Irish Deputy Prime Minister Simon Coveney said May planned to go to Strasbourg later Monday "to try to finalize an agreement, if that's possible".
"These resolutions will remain in place until the board determines that the ownership and control of the company is no longer such that there is any risk to the airline licences held by the company's subsidiaries pursuant to EU Regulation 1008/2008", Ryanair said.
"It is inevitable this unchanged withdrawal agreement will be voted down again", two senior Brexit-supporting MPs, Steve Baker and Nigel Dodds, warned on Sunday.
Talk of May stepping down as Prime Minister sooner rather than later is also growing, with her former policy adviser, Tory MP George Freeman, calling for "a new leader for a new generation" after Brexit.
"I think that an important offer has again been made to Britain, and now it is of course for Britain to respond to these offers", she said.
But May's spokesman, James Slack, said "talks are ongoing" at a technical level, and there is "a shared determination by both sides to find a solution".
If parliament does not support leaving the European Union without a deal, the government has promised to hold a further vote, asking lawmakers if they want the government to ask for a "short, limited" delay Britain's European Union exit.
So far, there's been widespread skepticism that she can win enough votes to push it through - her previous version of the plan lost by a stunning 230 votes in January. If that happens, Parliament remains gridlocked, the European Union reluctantly agrees to a short extension to avoid the pain of no-deal, and in three months time we're back to the cliff-edge.
That said, Labour looks set to abstain from putting forward an amendment in that direction this week, but individual MPs may decide to do so.
This would keep Britain in the EU's customs union and parts of its single market until and unless another way - such as a trade deal - is found to avoid frontier checks.
What's the likelihood Parliament would vote for a "no-deal Brexit"?
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.
For all of the prime minister's weaknesses as a politician - her repetition of talking points has earned her the nickname "the Maybot" - she has proved remarkably durable. She survived a bid to oust her through a no-confidence vote in December.