Unionist leaders will today tell the European Union's chief Brexit negotiator Michel Barnier that they will not accept Northern Ireland being treated differently from the rest of the UK.
UP leader Arlene Foster said that any barrier to trade between Northern Ireland and Great Britain would be "catastrophic", while Ulster Unionist leader Robin Swann said it would be as economically "harmful as a no-deal Brexit".
The talks come ahead of ahead of next week's EU summit, which is seen as crucial to the outcome of the Brexit negotiations.
Leaders of the 27 remaining EU states are due to gather in Prime Minister Theresa May's absence on October 17 for an eve-of-summit briefing from Mr Barnier at which he is expected to deliver his assessment of whether a deal is within reach.
Hopes that a withdrawal deal can be completed within weeks have been fuelled by weekend comments from Ireland's deputy prime minister Simon Coveney, who has insisted that both sides were 90% there.
Mrs Foster's meeting comes after comments last week that her party would not agree to any solution which would see a 'sea border' between Northern Ireland and the rest of the UK.
Speaking in advance of the visit, Mrs Foster said the DUP's only red line was that no new borders would be created between Northern Ireland and the rest of the UK.
"To create a barrier to that trade would be catastrophic," she said.
"We want to see an exit deal which means Northern Ireland has unfettered access to and from the GB market but also full beneficiaries of any new trade deals with the United Kingdom after Brexit.
"Our red line also respects and protects Northern Ireland's constitutional place in the United Kingdom. Many who claim to respect the Belfast Agreement fail to respect the principle of consent which was part of that agreement.
"Indeed, they would happily redraw the border and annex Northern Ireland away from the rest of the UK."
Meanwhile, Mr Swann said the legitimate concerns of unionists must not be dismissed in the Brexit negotiations.
"As unionists it is vital to us that the constitutional integrity of the United Kingdom is preserved throughout this process, but it is also clear that in terms of the economic well-being of Northern Ireland, any border along the Irish Sea would be as harmful as a no-deal Brexit."
Pro-Remain parties - the SDLP, Sinn Fein, Green Party and Alliance - met Mr Barnier on Friday.
During a post-meeting press conference, SDLP leader Colum Eastwood and Sinn Fein deputy leader Michelle O'Neill said that giving the suspended Assembly any say over the backstop would be unacceptable.
Consulting Stormont on the backstop could effectively give the DUP a veto on a deal, they added.