Jeffrey Donaldson MP, who represents Lagan Valley, responded to a tweet from deputy prime minister of Ireland Simon Coveney which rejected a time-limited backstop or a backstop that could be unilaterally ended by the United Kingdom without the agreement of the Republic and the European Union.
At a weekly cabinet meeting, May told ministers she expected an agreement and wanted it as soon as possible, but it would "not be done at any cost", according to her spokesman.
Mr Johnson dismissed suggestions that the arrangement would lead the way to a Canada-style trade deal, arguing that the joint UK/EU political declaration on the future relationship due to be released alongside the withdrawal agreement will be "worthless".
The crunch meeting comes amid mounting pressure to finalise the withdrawal agreement in order to announce the leaving terms at a special summit in November.
But a spokesman for Mrs May said no extra meeting has yet been scheduled ahead of the regular cabinet meeting in Downing Street next Tuesday.
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.
"While the United Kingdom should aim to conclude the Withdrawal Agreement as soon as possible, this would not be done at any cost".
Ireland is willing to examine ways in which a backstop to keep the Irish border open after Brexit could be reviewed so long as it does not permit Britain to unilaterally walk away from it, Prime Minister Leo Varadkar had said on Monday.
The backstop plan enraged the DUP, which is propping up May's Conservative government at Westminster, and the PM has since said she could "not accept" any deal that would require a customs border between the North and Great Britain.
'All the messages that we get back over this whole period is that our European partners desperately want what we want - a deal that will protect their jobs and their economies in the same way that we want to.
The UK would effectively become a non-voting member of the European Union, forced to accept laws made in Brussels with no power to shape or amend them, he said.
"A choice between a no-deal Brexit and a no-clue Brexit is no choice at all".
The Daily Telegraph reported on Monday that Brexit secretary Dominic Raab had taken a "hardline" stance on the border, insisting that the United Kingdom should have the right to pull out of the backstop via a review process after just three months.
Looks like we're heading for no deal.