Britain's Prime Minister Theresa May is welcomed by European Commission President Jean-Claude Juncker at the EC headquarters in Brussels, Belgium December 4, 2017.
On Thursday, EU Commission spokesman Margaritis Schinas had dismissed British newspaper reports that the Sunday deadline could be extended into next week as "not correct". "What I said was that we would be willing to take a look at and consider any proposals that the United Kingdom side have".
He said no particular words or combination of words were discussed.
The political editor of Sky News, quoting a DUP source, said the party would continue to work on the border issue on Friday.
They meet in Brussels in a week to decide whether enough ground has been made to broaden the talks to future relations and trade, as Britain so badly wants.
Downing Street said Mrs May told Mr Varadkar she was "working hard to find a specific solution to the unique circumstances in Northern Ireland" and was committed to "moving together to achieve a positive result on this".
As speculation swirled about whether London, Dublin and Brussels were about to clinch a border deal, European Council President Donald Tusk's office said he would brief reporters at 0650 GMT on Friday before departing for Hungary.
The intense time pressure on the PM to get European Union agreement to move to trade talks will only grow with the Council summit looming large and the risk of trade talks being delayed until well into the New Year. May is struggling to balance those demands against the concerns of Northern Ireland's Democratic Unionist Party, which she relies on to support her government in Parliament.
A deal fell apart on Monday when the UK's pledge of "regulatory alignment" across Ireland - to avoid border customs checks and posts - was vetoed by the DUP.
It would also increase pressure from Tory Brexiteers for her to walk out of the talks and prepare for a "no deal Brexit".
Boris Johnson has urged the European Union to move Brexit talks forward so divisions over the Irish border can be "thrashed out".
"There will be areas where we do do things in a very similar way, there will be areas in which we don't do things in the a very similar way and that's all the PM was seeking to ensure - to make sure trade flows as freely as possible across the border".