Tajani added that May appeared neutral on the idea of extending the transition period by a year. She is hemmed in by pro-Brexit members of her Conservative Party, who oppose any more compromises with the bloc, and by her parliamentary allies in Northern Ireland's Democratic Unionist Party, who insist a solution can't include customs checks between Northern Ireland and the rest of the United Kingdom.
An offer of an extension clause on the 21-month transition period had already been made by the European Union during the recent secretive round of talks, as first revealed by the Guardian.
And former UKIP leader Nigel Farage spotted that the plan would take the country very close to the next election, due in June 2022. Despite her appearance, any hopes of a Brexit deal breakthrough is as low as it has ever been in recent weeks.
Critics claimed she was trying to "rig" the vote.
Leaders of the EU27 will discuss how to step up preparations for a no-deal Brexit next March at their working dinner in Mrs May's absence, following her presentation, he said.
Antonio Tajani, the European Parliament President, said: "I listened carefully to what May had to say".
Upon her arrival at the European Union summit, May told reporters, "We have solved most of the issues in the withdrawal agreement".
'As I see it the only source of hope for a deal for now is the good will and determination on both sides.
And she told them: "The last stage will need courage, trust and leadership on both sides".
"There were no new proposals".
Even if the prime minister gets everything she is demanding, and no one believes she will, Britain will have to continue abiding by European Union rules over which it will no longer have any say without the ability to pursue an independent trade policy while facing repeated negotiating cliff-edges for many years to come.
But Mr Barnier made clear no breakthrough was now expected, saying that "much more time" was needed to bridge differences between the two sides, and promising to "continue the work in the next weeks calmly and patiently".
Earlier, Irish Prime Minister Leo Varadkar said a longer transition period was not a substitute for an agreement on the so-called Irish backstop, created to prevent the need for customs checks at the border between Northern Ireland and the Irish Republic until a wider trade deal is done.
But Brussels feels the plan will need such complex policing and checking that it can not be sorted in time for agreement this month.
But that idea has been shot down by Europe's chief negotiator, Ireland's prime minister and members of her own party, who want the ability to do trade deals with the rest of the world. The House of Commons would then have little time to scrutinise the proposals.
The cabinet meeting yesterday did not consider a specific wording on any deal, nor make any decisions, but was a discussion about the negotiations and unity. "It will most definitely limit the prospects of our young", he said.
And it would be more than five years after the historic referendum vote to Leave.
Tory veteran Ken Clarke told the BBC's Newsnight: "This is the latest silly tactic, which they won't get away with. Parliament is going to probably want to give some detailed instructions for the government and temper its approval with conditions".
But Keir Starmer, the Brexit spokesman for the opposition Labour party, said that lawmakers should be able to scrutinize and amend any government proposal.
What is the meaningful vote?
Protesters will begin assembling from midday on Saturday on Park Lane in central London before descending onto Parliament Square where speakers will deliver in support of a People's Vote from 2pm onwards.
This offer would not in itself resolve the back-stop issue, which must be settled in the Brexit treaty that must be ratified before March to avoid a damaging "no deal" scenario.