Tusk said in October that if London failed to settle divorce terms by the end of this year, the EU would have to reconsider its approach to negotiations meant to limit disruption when Britain leaves the European Union in March 2019.
The pro-British DUP party which props up May's government refused Monday's draft deal over a phrase suggesting there would be "regulatory alignment" with the European Union after Brexit - effectively putting Northern Ireland under European Union law.
DUP leader Arlene Foster and Theresa May reportedly spent hours on the phone overnight discussing the post-Brexit Irish land border.
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.
An agreement on the Irish border was scuttled at the last minute Monday by Northern Ireland's Democratic Unionist Party.
"While we support David Davis' ambition to secure a comprehensive agreement by 29 March 2019, nearly nobody outside the Government thinks this will be possible", Jay said.
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Barnier informed European Union ambassadors that Downing Street had told him a potential solution was being worked out that could possibly satisfy both Northern Ireland's Democratic Unionist Party (DUP) and Ireland, but that it had yet to be signed off by any of those involved, the Guardian reported on Thursday. "We've seen a rally in the pound over the last couple of weeks on the expectation of a deal being agreed so what we may be seeing is simply a case of buying the rumour and selling the fact", he said.
The announcement that European Council president Donald Tusk will make a statement on Brexit early on Friday morning added to talk that the Prime Minister had made progress.
But it doesn't all end in March 2019, when the U.K.is slated to exit.
The DUP insists it will not accept any agreement in which Northern Ireland was treated differently from the rest of the UK. "In Northern Ireland we guarantee there will be no hard border", May told a press conference with Juncker.
Mrs May has said she wants a deep and special partnership and a better deal than the free-trade agreement that Canada secured from the EU.
Mrs May's Conservative administration is fiercely divided over Brexit - her Cabinet has yet to decide what kind of trading arrangements it wants from Europe.