The EU executive said a compromise had been reached after tough talks that would allow the broadening of Brexit talks to include issues of future trade relations.
Talks between May and Juncker in Brussels on Monday broke up without a deal after the pro-British DUP party in Northern Ireland that props up the British leader's government objected to a clause in the draft agreement about future arrangements for the Irish border.
A new round of talks between Brussels and London is relatively Brexit is expected today.
May and European Commission President Jean-Claude Juncker could meet early on Friday to seal a border deal, the European Commission's chief spokesman said.
The Republic of Ireland, on the other hand, which is an European Union member, wants a guarantee that there will be no hard border between it and Northern Ireland after Brexit.
On Thursday evening, European Commission spokesman Margaritis Schinas tweeted: "We are making progress, but not yet fully there", adding: "Tonight more than ever, stay tuned".
The long-awaited announcement also followed talks which continued into the early hours between the Prime Minister and Democratic Unionist Party, Arlene Foster.
"We believe there is still more work to be done to improve the paper".
But the EU will only move to trade talks if there is enough progress on three key issues: the money Britain must pay to the EU; rights for EU citizens in Britain and British citizens in the EU; and how to avoid a hard border with Ireland. If a hard Brexit for most of the United Kingdom and a soft Brexit for Northern Ireland is out of the running because of the DUP, and a hard Brexit for the whole of the United Kingdom including Northern Ireland is out because of the border issue, that leaves only a soft Brexit for all of the UK.
May's office said she and Varadkar are "working hard to find a specific solution to the unique circumstances in Northern Ireland", which have triggered a crisis in Britain's divorce negotiations.
The agreement paves the way for European Union leaders at a summit on December 14-15 to open the second phase of Brexit negotiations, covering trade talks and a transition period.
But if a deal is to be in place and approved before the March 2019 Brexit deadline, negotiators probably have less than a year, Tusk said.
The DUP insists it will not accept any agreement in which Northern Ireland was treated differently from the rest of the UK.
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