EU Council President Donald Tusk will be in Dublin on Tuesday to meet with Taoiseach Leo Varadkar and it is expected the shape of a potential Brexit extension could be discussed at that meeting.
Over the past week, a series of Brexit votes have taken place in the Commons.
Unless May can get a Brexit deal approved by the British parliament, then she will have to decide whether to delay or cancel Brexit or thrust the world's fifth largest economy into chaos by leaving without a deal.
The Northern Irish party crucial to UK Prime Minister Theresa May's hopes of getting her twice-defeated Brexit deal through parliament said it had good talks with ministers but differences remained over the Irish border.
Mrs May says she wants to minimise any delay to just three months, but to achieve that she will need parliament to back her deal at the third time of asking early next week.
"We've always said in Ireland, if the United Kingdom asks for more time, for one to prevent no deal, a crash-out Brexit happening, but secondly to implement a clear plan and strategy to reflect on and perhaps change direction in regard to Brexit, well then Ireland certainly wouldn't be an obstacle to that", Coveney told a conference.
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The main motion was backed by MPs from across the various political parties, however most Conservative MPs voted against the motion. The U.K. will lose membership in the European Union institutions as soon as the transition period kicks in, but will need to abide by European Union rules till the transition is over. "As a delay was passed by Parliament, I want to see deal agreed ASAP so we can minimize to short, technical, extension".
Downing Street said this was a "natural consequence" of Mrs May's decision to offer a free vote on an issue where there are "strong views on all sides of the debate". He said there were now two options: "To vote for the deal and leave in orderly way or a long delay and I think that would be a disaster".
The march has cross-party support, with Labour and Conservative MPs joining, as well as members of the newly formed Brexit Party. Labour MP Ruth Smeeth resigned her position as parliamentary private secretary to Deputy Leader Tom Watson after voting against a second referendum. If the extra time is only a few weeks, the challenge is to come up with another deal acceptable to both British lawmakers and the EU.
Why did MPs oppose the deal?
"It's not up to European Union countries to say what that is, but the United Kingdom", it said.