The Irish Prime Minister said he was "surprised and disappointed" that the agreement - which would have seen the Republic and Northern Ireland maintain "regulatory alignment" after Brexit to avoid the return of customs posts - had collapsed.
Early on Monday, the UK government appeared confident that May would broker a deal at a Brussels lunch with European Commission president Jean-Claude Juncker and EU chief negotiator Michel Barnier.
A deal would have enabled the two sides to finally move on to trade talks in the New Year.
But after Irish state broadcaster RTE reported that May was set to concede to Dublin demands over the border, Arlene Foster, the leader of Northern Ireland’s Democratic Unionist Party (DUP) - whose MPs prop up May’s minority government - made it clear that her party would reject the proposal.
Foster also made her feelings known to May during a hastily-arranged phone call midway through her lunch with Juncker.
Making clear his own frustrations, Varadkar said: "Following the government meeting this morning, the Irish negotiating team received confirmation from the British government and the Barnier task force that the UK had agreed a text on the border that met our concerns. This text would form parts of the broader UK-EU agreement on phase one (of the negotiations) and would allow us all to move onto the phase two.
"I was then contacted by President Juncker and President Tusk and I confirmed to them both Ireland's agreement to that text. I am surprised and disappointed that the British government now appears not to be in a position to conclude what was agreed earlier today.
"I accept that the Prime Minister has asked for more time and I know that she faces many challenges and I acknowledge that she is negotiating in good faith but my position and that of the Irish government is unequivocal" Irish PM Leo Varadkar
"I accept that the Prime Minister has asked for more time and I know that she faces many challenges and I acknowledge that she is negotiating in good faith but my position and that of the Irish government is unequivocal."
Despite the setback, May insisted a deal could be struck by the end of the week.
Addressing journalists at a joint press conference with Juncker, May said, "We’ve been negotiating hard and a lot of progress has been made and on many of the issues there is a common understanding and it's clear, crucially, that we want to move forward together.
"But on a couple of issues some differences do remain which require further negotiation and consultation, and those will continue but we will reconvene before the end of the week and I’m also confident that we will conclude this positively."
Juncker commented: "Despite our best efforts and the significant process we and our teams have made in the past days on the remaining withdrawal issues, it was not possible to reach a complete agreement today,"
"On a couple of issues some differences do remain which require further negotiation and consultation, and those will continue but we will reconvene before the end of the week and I’m also confident that we will conclude this positively" British PM Theresa May
“We now have a common understanding on most relevant issues, with just two or three open for discussion. “These will require further consultation, further negotiation and further discussions.
The Commission president, however remained upbeat that a deal could be salvaged in time for a crunch European summit in mid-December.
“We stand ready to resume the negotiations with the United Kingdom here in Brussels later this week. “I’m still confident that we can reach sufficient progress before the European Council of December 15.
“This is not a failure, this is the start of the very last round. I’m very confident that we will reach an agreement in the course of this week.”