Irish Prime Minister Leo Varadkar, left, shakes hands with European Council President Donald Tusk after making a joint statement following their meeting at the Europa building in Brussels, Wednesday, Feb. 6.
The only Northern Irish party represented in the House of Commons, the Democratic Unionist Party, reiterated its opposition to the backstop, which is created to prevent a post-Brexit hard border on the island of Ireland.
While Tusk was clear the European Union would not reopen the divorce deal, he also said he still believed that a common Brexit solution was possible.
And at the end of their press conference, Mr Varadkar was picked up by the microphones telling Mr Tusk: "They'll give you awful trouble in the British press for that".
The comment came after Irish prime minister Leo Varadkar was caught on camera telling them that he will get a "terrible trouble" from the press for his strong remarks on Brexit.
"We will not gamble with peace or put a sell-by date on reconciliation".
Sammy Wilson, the Democratic Unionist Party's (DUP) Brexit spokesman, said, "It is Tusk and his arrogant European Union negotiators who have fanned the flames of fear in an attempt to try and overturn the result of the referendum".
The so-called backstop is created to preserve the open border between Northern Ireland and EU member state Ireland in light of the fact that May has insisted that the United Kingdom does not remain either in the European single market or the customs union.
"The top priority for us, remains the issue of the border on the island of Ireland, and the guarantee to maintain the peace process in accordance with the Good Friday Agreement". The E.U.is first and foremost a peace project.
The Irish prime minister again insisted that there were no "alternative arrangements" to the already agreed Northern Ireland backstop.
The Brexit chief for Northern Ireland's Democratic Unionist Party, Sammy Wilson, branded Tusk a "devilish, trident wielding, euro maniac". May is holding talks with the five main parties at Stormont House.
"Parliament voted by a significant margin to set out what it wishes to achieve, which is changes to the backstop", the Number 10 spokesman told reporters.
Neither will the deal's "backstop" that ensures an open Irish border, he said, denouncing the British eurosceptics now pushing for an abrupt "no deal" departure.
Critics of the backstop argue its lack of any agreed time-limit is unacceptable as it could see the United Kingdom locked into a customs union deal with the EU indefinitely and Northern Ireland kept under EU single market rules.