Secretary of State Theresa Villiers has said a referendum on a united Ireland would not be “justified or helpful” at present.
Speaking at Westminster, Ms Villiers said deputy First Minister Martin McGuinness had requested a referendum on a united Ireland.
He was reacting to Northern Ireland's vote to remain in the EU, which was in contrast with the UK's overall vote to leave.
“I do not believe that such a referendum would be justified or helpful at present,” she commented.
“The Belfast Agreement makes clear that the conditions under which I would be required to hold such a referendum.
“Those are that I have reason to believe there might be a majority support for a united Ireland.”
Speaking at Stormont Castle, Mr McGuinness said Irish premier Enda Kenny needs to support Northern Ireland in the aftermath of the referendum.
“What the Taoiseach needs to understand is that we do believe that there is now against the backdrop of this decision, which has been so detrimental to everybody on this island, for a border poll,” he commented.
“People say when we raise the issue, would this not cause instability, would it not lead to a very fractious situation. Why would it?
“I believe that such a border poll and the run into it can be conducted in a civilised atmosphere, just as the Scottish referendum was.”
DUP leader and First Minister Arlene Foster said Sinn Féin’s border poll call was “as predictable as the flowers in May”.
“We knew it would come but the test has not been met and therefore I do not believe it will happen,” she commented.
Ulster Unionist leader Mike Nesbitt had earlier said he was disappointed by the nationalist call, claiming it would introduce further uncertainty following the referendum outcome.