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.
The question of how to maintain a soft Irish border had emerged as the key sticking point to getting agreement from the European Union to move on to phase two in the Brexit negotiations.
Despite intense efforts over the weekend to agree on a proposal on how to avoid a hard border in Ireland, Irish officials revealed on Sunday night that "there is still a way to go" to achieve a meeting of minds on the issue, reports the Guardian. "It was reported as if it was true, and now it turns out it was propaganda from the Irish Government", he said.
There has been "some progress" towards a Brexit agreement on the Irish border but it remains "50/50" whether a deal will be reached, Sky News understands.
Mrs May hopes to achieve a breakthrough during her visit to Brussels to meet Mr Juncker and European Union chief negotiator Michel Barnier on Monday.
The border issue remained the main sticking point. Officials said he was preparing to call round European Union leaders to get agreement on trade negotiations.
May said that "on a couple of issues some differences do remain which require further negotiation and consultation".
Nassar victims demand answers from Michigan State University, USA Gymnastics
Separately, Nassar is charged with similar crimes in Eaton County, the location of an elite gymnastics club. He is facing similar charges in the MI area, which have been filed by more than 125 women and girls.
Juncker described May as a "tough negotiator", and insisted that talks had not failed.
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 now have a common understanding on most related issues with just two or three open for discussion".
"Not every single question has to be answered but we need sufficient progress on these very sensitive issues". But then Northern Ireland's Democratic Unionist Party, which props up May's minority government, announced it wouldn't support any deal that made special rules for Northern Ireland.
"We will not accept any form of regulatory divergence which separates Northern Ireland economically or politically from the rest of the United Kingdom", she said.
Varadkar said he was "surprised and disappointed that the British government now appears not to be in a position to conclude what was agreed earlier today". "If no clear commitment is made, the EPP group will not be ready to assess the progress made as sufficient to enter a 2nd phase of negotiations", Manfred Weber, German MEP and leader of the largest group, the European People's Party in the EP, tweeted.