Taoiseach Enda Kenny has said that he has held useful discussions with the European Commission's chief Brexit negotiator, Michel Barnier.
Mr Barnier told reporters in Government Buildings that he was very aware of Irish concerns and that he was ready to find solutions
Mr Kenny earlier said the challenges arising from Brexit were extremely serious and it was essential that all affected have an opportunity to have their voices heard.
He was speaking after Mr Barnier addressed a joint sitting of the Dáil and Seanad.
The Taoiseach thanked Mr Barnier for his engagement with Ireland since he became head of the commission's Brexit taskforce.
He said there was a political challenge ahead and there would be need for a "flexible and imaginative" approach.
Mr Kenny welcomed that the unique circumstances that apply to Ireland were fully acknowledged in the Commission's guidelines in supporting and protecting the achievements of the Good Friday Agreement.
However, he said it was not the time for a border poll.
He said Ireland's participation and membership of the EU had transformed the country over the past 40 years and it had played a strong and active role and would ensure its concerns and priorities are reflected as the union evolves.
Brexit not a positive assertion of sovereignty - Martin
Fianna Fáil leader Mícheál Martin has told the Dáil that last year's Brexit referendum was "an ugly and negative affair" and "no amount of warm words and earnest statements can cover this up."
"There was no strategy for implementing Brexit, there was just a strategy for winning the vote through a combination of bluster and aggression.
He made the comments following Mr Barnier's address to the Oireachtas.
"It was not a positive assertion of sovereignty; it was the culmination of 30 years of an increasingly corrosive scapegoating of Europe and immigrants for the home-grown divisions in British society," Mr Martin said.
Fundamentally, the narrow Brexit majority represented a rejection of strong rule-based co-operation between states, he said.
It asserted a narrow vision of sovereignty that developed in the 19th Century and directly led to the two bloodiest wars in history, Mr Martin added.
"Let there be no doubt about where Ireland stands. We want nothing to do with a backward-looking idea of sovereignty.
"We remain absolutely committed to the ideals of the European Union.
"We see the union for what it is - the most successful international organisation in world history.
"The union is flawed, but its successes are undeniable."
Adams in call for special designated status for Northern Ireland
Sinn Féin leader Gerry Adams used his address to Mr Barnier to look for Special Designated Status within the EU for Northern Ireland.
He began with criticism of the bloc by claiming that "the EU is wedded to neo-liberal policies".
"These have created widespread hardship as austerity, deregulation and privatisation have undermined the social function of states and the rights of citizens, including the rights of workers.
"Increasingly people across the EU are uncomfortable with its direction.
"This has assisted the growth of far-right parties which exploit people's fears. Brexit is a consequence of that."
He said it is vital that the challenges of Brexit are met on an all-island basis.
Designated special status for Northern Ireland within the European Union is not about a hard or a soft Brexit but is about the best interests of the economy, the peace process and the people, Mr Adams said.
EU not trusted to do deal with UK - Boyd Barrett
Soldarity/People Before Profit TD Richard Boyd Barrett asked Mr Barnier for a guarantee that there would be no hard border post-Brexit and that the Irish people would be allowed vote on the final EU-UK Brexit deal.
He asked if Mr Barnier can be trusted with issues of international solidarity or the free movement of people. and he spoke of the number of refugees drowning in the Mediterranean Sea as they flee Africa.
Mr Boyd Barrett said that he does not trust the EU to do a deal with the UK and he asked for a guarantee that Europe will not try to break up the free travel area between Ireland and the UK and not implement a hard border.
Meanwhile, Labour leader Brendan Howlin said the Brexit negotiating guidelines do not "sufficiently recognise the unique challenges" facing the Republic of Ireland.
Brendan Howlin says the Brexit negotiating guidelines do not 'sufficiently recognise the unique challenges' facing the Republic of Ireland pic.twitter.com/Q5K7ztimQV— RTÉ News (@rtenews) May 11, 2017
He said that Brexit means the idea of achieving the European single market has been set back a generation or more.
"Bluntly, once the UK leaves, it will no longer make any real, practical, day-to-day sense to talk about our membership of a single market in relation to the goods and services that we import and export", he said.
Mr Howlin said "talk of the single market will, from our perspective, revert from being something approaching reality, towards something more closely resembling a pious aspiration."