Unionists yesterday rubbished claims that a no-deal Brexit would make a border poll more likely in the near future.
The BBC reported that one Cabinet minister warned that the Government "risked sleepwalking into a border poll", while another felt a vote on taking Northern Ireland out of the Union was "a real possibility" if the UK leaves without a formal deal with the EU next month.
But DUP leader Arlene Foster, speaking after a meeting with Taoiseach Leo Varadkar, described border poll speculation as "Project Fear" propaganda, aimed at putting pressure on MPs to back Prime Minister Theresa May's unpopular withdrawal agreement, which she is attempting to renegotiate with EU leaders after it was rejected by an historic majority in the House of Commons
"There are many people engaging in Project Fear at this point in time and we all have to recognise that," the Fermanagh MLA said.
"The Belfast Agreement sets out the criteria for a border poll and it hasn't been satisfied and therefore will not be called."
Ulster Unionist MLA Doug Beattie also dismissed the speculation about a prospective border poll.
"Even without the upheaval of the Brexit process, such a poll would inevitably lead to instability and polarisation, and only serve the interests of those who want to hold Northern Ireland back," he said.
"If Cabinet ministers are seriously raising a border poll as an issue, they should talk to their colleague the Secretary of State for Northern Ireland, because the relevant legislation states quite clearly that it is her decision as to whether or not a poll should be called.
"As the Ulster Unionist Party has repeatedly said, there is no need for a border poll.
"The Secretary of State has consistently said that the conditions for calling a border poll have not been met, so these unnamed Cabinet ministers should settle themselves."
Pressure for a border poll has been coming from republican and nationalist politicians, with Sinn Fein demanding moves to put one in place.
Yesterday Sinn Fein vice president Michelle O'Neill said she had once again raised the issue with Mr Varadkar.
"We have put the issue of a unity referendum to the Taoiseach, to the British Prime Minister on every occasion on which we would meet them," she said.
"Remember, the unity referendum is built into the Good Friday Agreement, it will be for the people of this island to decide the constitutional future.
"Clearly, we want to see a deal. We do not want to see a crash-out Brexit.
"But if we do find ourselves in the scenario where there is a crash-out Brexit, then the tools which the British Prime Minister and Taoiseach must look to are actually written into the Good Friday Agreement and that is the unity referendum."
Mrs O'Neill said her party intended to keep the pressure on Mr Vadakar over the border poll issue.
"I think the Taoiseach has remained firm throughout the negotiation in recognising the need to protect Irish interests, in recognising the need to make sure that citizens in the north are never left behind," she added.
"He restated that commitment to us today.
"At the end of the day, we will hold the Taoiseach's feet to the fire on the issue."