DUP leader Arlene Foster has called for a "sensible Brexit" that "works for everybody".
Mrs Foster described her meeting with the new Taoiseach, Leo Varadkar, as "very positive", adding that she looked forward to a very positive relationship with the Irish premier.
The pair have previously worked together when Mr Varadkar was the Republic's tourism minister and Mrs Foster's Enterprise brief included responsibility for tourism.
Meanwhile the pair discussed the restoration of power sharing at Stormont, with the DUP leader insisting she was ready to talk to Sinn Féin.
"It takes two to tango and we're ready to dance," she said.
Northern Ireland has been without a power-sharing executive since March and without a first and deputy first minister since January, after Sinn Féin collapsed the administration amid faltering trust and relations with the DUP.
Talks to restore confidence took a back seat in recent days as the political focus largely shifted to London and the DUP's deal to prop up the Conservatives at Westminster.
But Stormont parties have a June 29 deadline to end the impasse and reach consensus on re-establishing a devolved administration in the region. "We want to see an administration set up again that will last and one that will last for all of the people of Northern Ireland," said Ms Foster outside Dublin's Government Buildings.
"We will go into speak with Sinn Féin again on Monday morning to try and get that set up as quickly as possible, because devolution works and works for everybody in Northern Ireland."
Ms Foster said the talks deadline remained "realistic".
"I think it is very much doable to have a deal by the end of this month," she added.
The DUP leader dismissed suggestions her deal with the Tories undermined the peace process.
"There is an irony to being lectured by some about our role in the national government of the United Kingdom when Sinn Féin want to be in government here in the Republic of Ireland," she said.
"What would happen then? Would we then say they shouldn't be in government in the Republic? They can't have it both ways, it has to be dealt with sensibly," she said.