May and European Commission President Jean-Claude Juncker fell short of a breakthrough, despite some encouraging progress on the Irish border issue.
European Union chief Donald Tusk said yesterday he was "encouraged" by progress in divorce talks with Britain and said that both sides were getting closer to the point where they can start focusing negotiations on future trade relations.
The EU has had "enough time now to decide whether or not they are going to discuss trade with us, they need to get on with it and if they don't get on with it, the closer we get to walking away with no deal", she said.
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.
"We had an agreement this morning", Irish Prime Minister Leo Varadkar said, expressing disappointment at the last-minute glitch.
David McAllister, a vice-president of the European People's Party - the European Parliament's largest grouping - said it was "difficult to predict" whether EU leaders will sanction the start of second phase Brexit talks this month, also classing the prospects of progress at "50/50".
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While Prime Minister Theresa May has previously insisted the entire United Kingdom will leave the single market, European Council president Donald Tusk made clear that Brexit talks cannot move on to trade issues unless the United Kingdom can satisfy Dublin there will be no return to a strictly-controlled border in Ireland. The lack of progress so far has raised concerns that Britain may not have a deal by the time it officially leaves on March 29, 2019.
Brussels officials and diplomats sounded increasingly confident of a deal over the weekend, but they caution that much will depend on the outcome of May's talks over lunch.
She said: "There are a couple of issues, some differences do remain which require further negotiation and consultation". But the issues of the rights of expatriate citizens and the UK-EU border on the island of Ireland defied a deal until the last minute. "We have common understanding on most issues".
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".
The Prime Minister is reliant on any support from the party as the DUP is propping up her government with its 10 MPs.
"I still hope this matter can be concluded in the coming days", he said.