Mr Coveney's spokesperson said the European Union had been united through the Brexit process and the UK had given written commitments that the Withdrawal Agreement would give a legal guarantee of no return to a hard Border in Ireland in any circumstance.
"Looks like we're heading for no deal", Jeffrey Donaldson, one of 10 Democratic Unionist Party (DUP) lawmakers whose support May now needs to get any deal passed in the British parliament, said on Twitter.
Last weekend, the London-based The Daily Telegraph reported that British chief negotiator for the Brexit, Dominic Raab, privately asked Coveney for the UK's right to unilaterally deactivate the emergency plan.
A spokesman for Ireland's Tanaiste Simon Coveney said: "The UK has".
Senior EU figures have indicated that they are prepared to offer British Prime Minister Theresa May an "independent mechanism" by which Britain could end a temporary customs arrangement with the bloc, the Times reported, without citing sources.
Theresa May is expected to brief her cabinet today on proposals to avoid a hard border in Ireland through a UK-wide customs arrangement that would eliminate most checks on goods. To ease Conservative fears that the United Kingdom could effectively stay in the EU customs union indefinitely, preventing trade deals with other countries, Downing Street is pushing for a review mechanism that would allow the United Kingdom to exit the arrangement.
May told Varadkar that there would need to be a mechanism through which the backstop could be brought to an end, a spokesman from her office said in a statement.
A spokesperson said: "The prime minister has been clear that we are making good progress on the future relationship and 95pc of the Withdrawal Agreement is now settled and negotiations are ongoing".
This could give mainland Britain some scope to set trade rules while keeping the province of Northern Ireland aligned with the EU. "That is why we are increasingly positive on the expectation of reaching a deal".
He argued that Britain would effectively become a non-voting member of the European Union, having to accept laws made in Brussels with no power to influence them.
The key idea would be that Great Britain and Northern Ireland would remain a single customs territory under World Trade Organization rules, linked in a customs union with Ireland and the rest of the EU, diplomats told Reuters.
This was to be a "backstop" insurance clause in the Brexit treaty, to be triggered if, by the end of a status-quo transition period, no better way had been agreed to keep the Irish border operating more or less as it does now.
Even if a deal is done in Brussels in the coming days, May will have to sell it in London - first to her own Cabinet, and then to Parliament.