In a last-minute phone call, DUP leader Arlene Foster threatened to pull her party's support for Mrs May's Government claiming the deal would have created unacceptable regulatory divergence between Northern Ireland and Britain.
The DUP scuppered this deal on the basis that it would move Northern Ireland closer to the Republic in economic terms.
Irish Taoiseach Leo Varadkar said that Theresa May has promised to come forward with new proposals for a deal on the Northern Irish border.
Pressure is growing on Mrs May to get leaders at the December 14 European Council summit to declare sufficient progress has been made on divorce issues so trade talks can begin, with business chiefs warning companies will activate contingency plans that will cost Britain jobs if there is further delay.
Dublin would look at new proposals but its core position needed to remain intact, said Mr Coveney. Ms May and Mr Varadkar spoke by telephone on Wednesday afternoon in an effort to break the impasse in the talks. "Work will continue in London".
The Republic insists that keeping the United Kingdom in the single market and customs union is the best way to avoid the return of a hard border.
He said that the "regulatory alignment" proposed by the British prime minister in Monday's text did not involve laws within the post-Brexit UK being the same as those in the rest of Europe.
Phase two of the Brexit talks will focus on the transitional arrangements that will kick in after the UK formally leaves the EU in 2019 and the future EU-UK trading relationship.
Duckett poured drink over senior England player, Bayliss confirms
After the Bairstow incident, director of cricket Andrew Strauss denied the England team had a drinking culture. Players were then given a midnight curfew which was only lifted on Thursday.
He added that the "absolute red line" that has always been there remains, and he is determined not to return to a hard border with Northern Ireland.
Answering questions after a speech at the Foreign Office in London, Foreign Secretary Boris Johnson declined to comment on the progress of Brexit negotiations, but said it was time for the European Union and Britain to "get going" with the second phase of talks. On this point, the ball is very much in the EU's court.
But he added, breaking briefly into French: "We need to get going, franchement (frankly), with the second part of the talks". We would have to ensure that any new wording is consistent with that.
Earlier, Britain's Foreign Secretary Boris Johnson insisted any deal must respect the promises made during the Brexit referendum campaign.
A majority of British voters decided 18 months ago to leave the European Union, but this first phase of talks - focused on preliminary issues that many had expected to be resolved quickly - has dragged on amid disputes over the divorce bill and the Irish border.
"She wants to come back to us with some text tonight and tomorrow", he said.
"I was asked my reaction to some of the very extortionate sums that I had heard in the region of £80 or £100 billion, and, I don't want to repeat the offending phrase, but go whistle seems the appropriate reaction to that kind of money".