According to a leak to the Irish state broadcaster RTE, the United Kingdom was willing to agree to Northern Ireland having "continued regulatory alignment" with the Republic of Ireland to ensure there was a continued soft border between the North and South.
The British government says that when it leaves the EU in March 2019, it will also leave the single market and the customs union.
Speaking at a joint press conference in Brussels with Mr Juncker, Mrs May said: "We have been negotiating hard".
All parties initially appeared to be confident that there was an agreement to be had on citizens rights, the financial settlement and the increasingly thorny issue of the Irish border, but the bullish talk came to nothing.
But they added: "A deal is sought, but it all turns on Ireland, and if the Irish government gives Barnier the green light".
There was even talk of Mrs May's set-piece lunch with Jean-Claude Juncker, the European Commission President, being used to thrash out the terms of a trade deal, with the terms of the divorce deal already in the bag.
The Irish prime minister said he is "surprised and disappointed" that there was no agreement in Brexit talks today.
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She also said that when women work, it creates a "multiplier effect", and leads to more reinvestment in families and society. Her India visit has made headlines but questions have been raised if she is the right person to talk about empowering women.
Britain is set to leave the European Union on March 29, 2019 yet the progress on its exit and the terms of new relations have been painfully slow.
If a Brexit deal can be done that "effectively" keeps Northern Ireland in the single European market, there is "surely no good practical reason" why Scotland should not benefit from such an arrangement, the First Minister has said.
It is understood this is a sticking point in reaching a deal on post-Brexit citizens' rights, the part of the phase one divorce negotiations that had previously been considered to be the most advanced.
If she had ignored their concerns, there's little doubt that the party's 10 MPs would have sat on their hands and not supported the Conservatives in important votes on the EU Withdrawal Bill this week.
"It is now getting very tight but agreement at December (summit) is still possible", he tweeted.
Weber said in a tweet that even if the issue of the outstanding bill had made major progress, "we are much more concerned about the fact that negotiations are stalled on the protection of European Union citizens' rights & on the Irish case".
"And the problem between the Republic and Northern Ireland is a effect of the British decision not only to leave the EU but also the single market and the customs union".
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