"But I would like to repeat what President Juncker said on this, namely that this will have to happen this week, in this building".
The Prime Minister is hoping to make a new offer by Friday to satisfy both Ireland and the Northern Irish Democratic Unionist Party which props up her Government and scuppered a deal on divorce issues on Monday.
"We are making progress but not yet fully there", Commission spokesman Margaritis Schinas said. "Early morning meeting possible". "We are here to work constructively with the United Kingdom government to come to an agreement based on the principles that we had agreed to yesterday", Varadkar told parliament on Tuesday.
Separately, Director of the Centre for European Reform Charles Grant said he believes the recent setback on the border issue has weakened London's position and set off reverberations in Scotland, which will now seek a similar deal to Northern Ireland.
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.
Amid signals Britain could seek "regulatory alignment" with the European Union to ensure there does not need to be a hard border, Foreign Secretary Boris Johnson insisted any Brexit deal must mean "taking back control of our laws, of our borders and of our cash".
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.
Michel Barnier put pressure on the Prime Minister by insisting diplomats of the 27 member states must sign off any potential agreement on Friday, if the talks are to move onto trade at a summit next week.
All sides say they want to avoid a return to a hard border between European Union member Ireland and the British-ruled province of Northern Ireland, which might upset the peace established after decades of violence.
May had came close to a deal on Monday, but it fell apart at the last minute over the Irish border.
DUP leader Arlene Foster speaks at her party's annual conference in Belfast, Northern Ireland, November 25, 2017.
"In Northern Ireland we guarantee there will be no hard border", May told a press conference with Juncker.