Mr Varadkar and Ms May spoke earlier on the phone, and the Taoiseach said he would look at positions proposed by the UK.
Minister Coveney noted that both sides are facing into "a sensitive negotiation at a very sensitive time" and insisted the Government would not say anything to "stoke up what is already a hard relationship management exercise".
A DUP spokesman said: "There is still plenty of work to be done".
British Prime Minister Theresa May told her Irish counterpart Leo Varadkar that she would propose suggestions to Brexit negotiators over the next 24 hours to try to break an impasse on the issue of the Irish border, Varadkar said on Wednesday. In a referendum on June 23, 2016, about 51.9 percent of British said "yes" to their country leaving the bloc.
This morning Transport Secretary Chris Grayling, who is one of the most prominent Brexiteers in the Cabinet, said: "I remain absolutely optimistic that we will reach a successful point, we will move on to the trade talks, because ultimately it is in everybody's interests for that to happen".
Canning Vale woman faces court charged with murdering Baldivis teen
No bail application was made but Ms Hishmeh was told it was her right to apply for bail in the Supreme Court at any stage. Ms Hishmeh appeared unable to stand on her own as she was led out of the court room, still sobbing, by two guards.
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.
British Prime Minister Theresa May on Wednesday voiced confidence in striking a deal that would allow Britain to open negotiations on post-Brexit trade ties with the European Union despite this week's setback.
As well as trying to appease the DUP, Dublin and Brussels, Theresa May also needs Conservative MPs to back whatever solution she puts forward. Arlene Foster's party was backed by Eurosceptic members of the Tory party.
But they hit out at what they say are attempts by some in their party to paint a no-deal scenario in which the United Kingdom failed to agree a trade agreement as "some status quo which the United Kingdom simply opts to adopt".
That's the suggestion from a House of Lords committee, which says a "time-limited extension of the UK's European Union membership" would "buy time" for a deal to be reached.
But the Brexit committee's chair, Hilary Benn, described the decision as "rather strange" since authorities hope to renegotiate of Britain's trade relations with the rest of Europe within weeks.
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