Barnier underscored the importance of settling the Irish border, saying "we can not afford to lose time on this issue and this is why we have invited the U.K.to work" on the issue next week.
With the prime minister opposed to anything that threatens the constitutional integrity of the United Kingdom, the impasse has become a major barrier to concluding a Brexit deal. Here are the main takeaways.
It forms part of a two-day visit to Northern Ireland focussing on Brexit and the stalemate at Stormont.
But, if implemented, it would effectively create a border down the Irish Sea between the island of Ireland and Great Britain.
Her bottom line, she will never accept such a proposal and believes no British Prime Minister could ever accept it.
The prime minister said a seamless border is "the foundation stone" of the peace agreement that ended decades of violence between Protestants and Catholics as well as between Britain and those wanting to unite Ireland with Northern Ireland.
The UK still needs to come up with a backstop that works.
"The situation at the moment is that the United Kingdom is part of the single European sky".
Mrs May has committed to ensuring there is no hard border with Ireland with no infrastructure, but it is yet to become clear how she can achieve this without being part of a customs union with the EU, something she is ruling out.
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European Union nations just as steadfastly insist on maintaining the spirit of the Good Friday agreement and also vehemently oppose any hard border on the island. "I think as you get closer to the line, the preparations need to be intensified".
What's Republic of Ireland's position?
Ahead of the meeting, deputy premier Simon Coveney, who will brief ministers about the recruitment of custom officers and plans for ports and airports, said the Irish Government was continuing to see "confusion" at Westminster following the publication of the White Paper last week.
Ahead of the visit, he posted on Twitter that "if the UK Govt (government) don't support current European Union wording on Backstop in draft Withdrawal Agreement, then obligation is on them to propose a viable and legally operable alternative wording that delivers same result: no border infrastructure".
Yesterday, Leo Varadkar, the Irish Taoiseach, said that the European Union has reassured the Irish Government "on many occasions" that "they wouldn't require us to put in place a physical infrastructure and customs checks on the Border between Northern Ireland and Ireland", even in what he termed the "doomsday scenario" of no Brexit deal.
Where do May's own MPs stand on this?
The Prime Minister's Chequers White Paper was criticised by Brexiteers in her own party, who bemoaned its strategy for a soft Brexit. It led to the resignation of several key members of her Cabinet including then-Foreign Secretary Boris Johnson and Brexit minister David Davis.
"And as they made clear this week, it is not something the House of Commons will accept either".