Sinn Féin president Gerry Adams has said he welcomes Micheál Martin’s support for an Irish border poll.
He said he has now written to the Taoiseach to “move this process forward”.
It follows on from comments made by the Fianna Fáil leader during a speech in Glenties, Co Donegal over the weekend.
Micheál Martin said Northern Ireland’s vote to ‘remain’ in the Brexit referendum could be a “defining moment” and “may show people the need to rethink current arrangements”.
He added: “I hope it moves us towards majority support for unification, and if it does we should trigger a reunification referendum.”
Reacting to the comments, Gerry Adams said: “I want to welcome Micheál Martin’s support for a potential referendum on Irish unity.
“It is incumbent on all of us who aspire to a united Ireland to become persuaders for a referendum, and I hope he will join with Sinn Féin and others in that endeavour.
“There is now an opening to examine new relationships for everyone who lives on the island of Ireland and we should seize that opportunity.
“I have written to the Taoiseach today to move this process forward.”
However the calls have been rejected by James Brokenshire, the newly-appointed Secretary of State for Northern Ireland, who was in Belfast on Monday.
Mr Brokenshire said it is difficult to see how Northern Ireland could remain in the European Union, following the Brexit referendum.
He also said he wants to make dealing with the past a priority.