At present the two sides say Britain will remain inside the European Union single market, and subject to the bloc's regulations, from the day it leaves on March 29 until December 2020, to give time for new trade relations to be set up.
Yet if a deal is to be struck, the focus will be finding a formulation that makes a customs border between Northern Ireland and the British mainland highly improbable and/or makes a UK-EU future customs zone more certain than Brussels now wants.
In the same breath, Mr Barnier admitted: "I'm convinced a deal is necessary, I'm still not sure we'll get one".
The extension idea has angered pro-Brexit U.K. politicians, who see it as an attempt to bind Britain to the bloc indefinitely.
In an open letter to May published on Thursday, leading Brexiteers accused the European Union of "bullying" and said the border issue was being used as "a trap" by the bloc.
It is adamant it will not agree to anything which results in imposition of customs or trade checks between Northern Ireland and the rest of the United Kingdom, known as a border in the Irish Sea. "The last stage will need courage, trust and leadership on both sides".
May urged both parties to show "courage, trust and leadership", but came to Brussels without the concrete new proposals the European Union has asked for.
The lack of progress means a special European Union summit on Brexit that had been penciled in for next month to finalize a deal has been scrapped, though European Union leaders said they would assess the situation in the coming weeks.
And one person familiar with the discussions said May's Tory party would find it hard to be fighting the next general election - due in 2022 - while the country is still inside the single market and customs union.
EU Council President Donald Tusk said the summit also left him more optimistic than he'd been after a bad-tempered meeting last month and a negotiating crisis on Sunday.
Both sides have suggested compromises, but this has still not brought an end to the deadlock.
May also said that the United Kingdom is ready to consider extending the transition period after Brexit day, in March 2019, to up to three years, from the current two, according to several officials.
"I believe a deal is achievable and now is the time to make it happen".
In Brussels, the expectation is if there is a deal, it won't be ready to be signed off until the European Council's December summit, leaving parliaments there and in Westminster just weeks to scrutinise and vote on it.
Sinn Féin's Michelle O'Neill said the prime minister was "ignoring majority voices" and SDLP leader Colum Eastwood said it was "disgraceful". European leaders are now discussing another summit in November - or even December before the holidays - to secure a possible accord.
Despite the lack of progress, the mood music at the summit was positive. This is now set to last 21 months, between the exit day on March 29 next year and the end of 2020. A deal must be sealed soon so parliaments have time to give their verdict on it.
"There's no need to dramatize matters".
"The blame for this mess falls squarely at the Prime Minister's feet", said the party's Brexit spokesman, Mr Tom Brake.
May also faces a rebellion from her parliamentary partners, Northern Ireland's Democratic Unionist Party, which has threatened to vote against her government's budget if she concedes to the EU's demands over the backstop. It will only be triggered if both sides do no reach a future trade deal.
The summit continues Thursday with an agenda limited to some issues both sides firmly agree on, including fighting cybercrime and dealing with an assertive Russian Federation. "The tone was of someone who wants to reach an agreement".
It has compounded her problems with a large rebel faction within her own party, however, and other hardcore Brexiters who fear negotiation delays and agreeing to keep Britain tied to the bloc for longer could mean the country, which is deeply divided over Brexit, ultimately never leaves.
"Yes, there are some hard issues that we are still working through", she said. They do not know themselves what they want.