Schinas said earlier on Monday that Brussels had offered London further reassurances on the temporary nature of the Irish border backstop. "It is now for the House of commons to make an important set of decisions this week".
May has secured "legally binding changes" which improve the Withdrawal Agreement and Political Declaration, Cabinet Office Minister David Lidington, who May's de facto deputy, told the British parliament.
If anything, the talks went backwards for May.
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 backstop, an emergency fix aimed at avoiding controls on the sensitive border between the British province of Northern Ireland and European Union member Ireland, was the most contentious part of the deal Mrs May agreed to in November.
However, the leader of the House of Commons, Andrea Leadsom, called the offer "disappointing", saying it simply brings back to the table a previously discarded option.
He said the United Kingdom could leave without a deal while negotiating the future trading arrangements.
Failure means Britain could end 46 years of ties with its closest trading partner on 29 March with no new arrangements in place, causing huge disruption on both sides of the Channel. However, the European Union has refused to back down on this measure, stressing the importance that keeping an open Irish border has on diffusing decades of violent tensions.
The impasse has raised fears of a chaotic "no-deal" Brexit that could mean major disruption for businesses and people in Britain and the 27 remaining European Union countries.
He said the government "absolutely stood by" its commitment to hold Tuesday's vote and, if the PM's deal was defeated, subsequent votes by Thursday at the latest on a no-deal exit and extending talks. There are many, even in the Prime Minister's circle, who now expect Brexit to be delayed beyond March 29. The first of those is scheduled for tomorrow, and appears to be heading for a defeat on the same scale May suffered last month. If that is passed another vote will take place to decide on delaying Brexit.
What will parliament vote on?
She has also been trying to win over Labour MPs from leave-voting constituencies with a 1.6 billion pounds ($2.1bn) fund and promises of more workers' rights. Brussels, which feels as if it has been asked to solve a domestic British political dispute, would want a clear explanation of how granting the United Kingdom several more months would solve an issue that has paralyzed Britain's political system.
British and European officials worked through the weekend to try to break the deadlock, and May is ready to make a last-minute visit to Brussels if necessary to seal the deal.
But the great majority of lawmakers, including most Conservative members of Parliament, will vote against a no-deal Brexit because they believe it would be economically damaging and disruptive.
If the deal is voted down once again, on the following two days the lawmakers are expected to discuss and vote on two amendments. Get ready for fireworks and for a lot of uncertainty when it comes - or if it comes at all.