The issue here is that, because of the May government's "red lines" of leaving the EU single market and customs union, her Brexit withdrawal deal contains a so-called backstop created to avoid a hard border (any kind of physical perimeter or visible customs checks) between the North and the rest of Ireland.
Still, May plans to meet with European Commission President Jean-Claude Juncker and other EU leaders in Brussels on Thursday seeking changes to the backstop, and will return to Parliament next week with - she hopes - a modified plan.
Dublin, he insisted, would not give up on an open-ended backstop obliging Britain to retain many EU rules until a better way could be found to keep goods flowing unchecked across the new EU-UK land border across the island of Ireland.
"We will find a way to deliver a Brexit that honors our commitments to Northern Ireland, commands broad support across the community in Northern Ireland and secures a majority in the Westminster Parliament", May will say, according to extracts of a speech released by her office.
Downing Street has indicated potential solutions could revolve around a time limit or unilateral break clause on the backstop or new technologies to make it unnecessary.
"The worst-case scenario is no deal Brexit and in our opinion we should do what we can to avoid that, to look for some solutions, to be creative", he told reporters in Brussels on Monday.
The EU will not renegotiate the divorce deal and the Irish backstop but alternatives could be worked on after Britain leaves, chief EU negotiator Michel Barnier said on Monday.
"More than ever this has demonstrated why we need to be part of an worldwide treaty that a prime minister, parliament or government of the day can not abrogate it".
Mr Varadkar also said Ireland is increasingly prepared for a no-deal Brexit.
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"We are now 51 days from the Brexit deadline and the British prime minister has come here empty handed with the same old rhetoric with no plan, no credibility and frankly no honour", said Sinn Féin's leader, Mary Lou McDonald.
Plans for Britain to leave the European Union on 29 March under a withdrawal agreement signed previous year were thrown into doubt when British lawmakers rejected the accord.
He added: "Alternative arrangements, they will never replace the backstop".
"The only way that anyone can act in a democratic fashion is to respect consent, to respect the democratic wishes of the people of the north of Ireland and to use the Good Friday Agreement, its provisions, to guide us in the time ahead".
There would also have to be no checks between Northern Ireland and the Irish Sea and Britain must be able to leave the alternative arrangement of its own accord, the MP said.
Despite this, the PM is adamant she will get changes made after being told by MPs in Parliament that her deal won't get passed unless she gets rid of the Irish backstop.
Theresa May - who supported the United Kingdom staying in the European Union during the 2016 European Union referendum - is due to arrive in Brussels on Thursday to seek legal changes to the withdrawal deal she signed with the EU.
May's government has been searching for a way to overcome lawmakers' opposition to the backstop, by adding a time limit or an exit clause - both of which have been rejected by the European Union - or a finding a way to replace border infrastructure with unspecified and untested technology.
A failure to move forward in February would mean increased prospects of an extension to the Article 50 process, which may only kick the can down the road, or the political and economic shock of a no-deal Brexit.