The Prime Minister today stressed to her Cabinet the importance of reaching an agreement this month but said this must "not be done at any cost".
"While we too hope the Northern Ireland backstop will never be required to be used, it will be required to be written down in legal text".
The EU's position to avoid a hard border has been for Northern Ireland to remain within the EU's customs union.
"He recalled the prior commitments made that the backstop must apply "unless and until" alternative arrangements are agreed." A Downing Street spokesman described Mrs May's conversation with Mr Varadkar as "constructive", adding: "They agreed that the intention was that the backstop should only be a temporary arrangement and that the best solution to the Northern Ireland border would be found by agreeing a future relationship between the United Kingdom and the EU".
Barnier also said a breakthrough on the Irish border issue was not close.
Pet owners will have to buy a £55 health certificate every time they go to Europe on holiday if Britain crashes out of the European Union without a Brexit deal.
Mr Barnier added a backstop arrangement "cannot have an end-date" and "must be applicable unless and until another solution is found".
May later reiterated to Austrian Chancellor Sebastian Kurz that she believed a withdrawal accord was 95% complete, and was "confident" of a deal on the Northern Irish backstop.
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Prime Minister Theresa May has previously stated she will not consider any proposals that would see Northern Ireland "carved out" of the United Kingdom.
Her optimism was immediately dampened by the EU's chief negotiator, Michel Barnier, who said that a deal on the Irish border was not "close" and that a "real point of divergence" on the issue remains.
"Deal or no deal, it's becoming clearer by the day that the United Kingdom is headed for a miserable Brexit".
But, highlighting the political complexities of Brexit, a large Survation poll for Channel 4 found British citizens would vote to stay in the European Union if there were another ballot, backing "Remain" by 54% to 46%.
May has repeatedly ruled out a referendum rerun, saying her job is to deliver the 2016 vote to leave the bloc, even as her plan has drawn critics from both sides of the Brexit divide.
May spokesperson said it was likely that a further cabinet meeting would be held before a deal was agreed, hinting that this could happen later in the week.
In response to tweet from Irish foreign minister Simon Coveney on Monday, the DUP's chief whip in the House of Commons, Jeffrey Donaldson, cautioned that it "Looks like we're heading for no-deal".
The conversation followed a report that Brexit Secretary Dominic Raab has privately demanded the right to pull Britain out of the Irish backstop after just three months.