Ireland, with the backing of the other 26 member states, wants the United Kingdom to provide guarantees on how to avoid a hard border between Northern Ireland - part of the United Kingdom - and the Republic of Ireland.
Giving a statement today, Taoiseach Leo Varadkar indicated he was still ready to support a deal but that the United Kingdom government may need more time to agree it.
A meeting on this with all political party leaders is set for lunchtime today.
Irish state broadcaster RTE said that a draft text would commit Britain to "continued regulatory alignment" with the European Union to avoid divergence on the two sides of the Irish border.
But critics say that it would effectively move the customs border between the United Kingdom and the Republic into the Irish Sea, if the rest of the United Kingdom was not in the single market.
But the two sides said they were within striking distance of consensus, setting up a hectic negotiating rush ahead of an European Union summit next week.
The border issue remained the main sticking point.
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Varadkar said he had spoken to Juncker following the failure to strike a deal "and he has confirmed to me that Ireland's position remains Europe's position".
The UK government has repeatedly insisted many of these questions can not be answered until the next stage of Brexit talks - on a transition period and a future EU-UK trading and customs relationship - is allowed to begin. Officials said he was preparing to call round European Union leaders to get agreement on trade negotiations. "But on a couple of issues some differences do remain which require further negotiation and consultation".
The EU and the United Kingdom are nearing agreement on some divorce terms, including the size of the bill that Britain must pay as it leaves and the rights of citizens affected by Brexit. "We have common understanding on most issues".
Ahead of the meeting, a source close to the discussions on the Irish border told Sky News: "There's been some progress on the wording from both sides, but a bit of a distance to go".
"We can not align the regulation of one part of the United Kingdom with the European Union", he said.
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.
He added: "I still hope that this matter can be concluded in the coming days as agreed".
"It is now getting very tight but agreement at December (summit) is still possible", he tweeted.
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