The Minister of State for European Affairs Helen McEntee has said it will be "very difficult" for Ireland to reconcile its obligations to the EU with protecting the Good Friday Agreement in the event of a no-deal Brexit.
But she said the Government is determined to avoid the reintroduction of checks on the border and has the support of the rest of the EU for that position.
"How we protect the single market and the customs union, these are discussions that we have been having and will continue to have into the coming week," she said.
She added: "It is about making sure that our obligations to the EU are fulfilled and the obligations we have as co-guarantors of an international peace treaty are also upheld.
"It is very difficult in the event of a no-deal to bring those two together. But we are absolutely determined to do that. We have always had the support of the EU and I don't see that changing."
Taoiseach Leo Varadkar is due to meet French President Emmanuel Macron on Tuesday and German Chancellor Angela Merkel on Thursday.
Ms McEntee said both meetings will involve discussions on the significance of a no-deal Brexit for the island of Ireland and both leaders will want to hear the Taoiseach's proposals for such a scenario.
She said Ireland favours a long extension to Article 50 which would delay Brexit.
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Fianna Fáil's spokesperson on Housing, Planning and Local Government has said a national discussion on Brexit must take place so people know what will happen in two weeks' time.
Dublin Fingal TD Darragh O'Brien has called on the Government to be up front about the situation, and questioned whether they are prepared for a no-deal scenario.
Mr O'Brien said he is still hopeful that a deal can be made, but that patience is wearing thin.
"We've had months and months of the Tánaiste and the Taoiseach saying they're hoping for a deal and that's still the likely outcome, but I think now the Government have got to be up front.
"They have to start talking publicly about what may happen and about what they're doing to lessen the blow that may be there."
Meanwhile, up to 100 people gathered on the border at Aghalane bridge which links counties Fermanagh and Cavan, as part of a series of protests against Brexit today.
Fr Joe Mc Veigh from Border Communties against Brexit called for people to make sure there won't ever be a hard border again on the island of Ireland. He was one of several speakers to address the crowd, among them local Cavan businessman Peter Quinn.
A few dozen protesters from Border Communities Against Brexit unfurled a banner outside Croke Park this afternoon as Leitrim fans gathered for a National Football League Final against Derry.
They said they wanted to show solidarity with the protests along the border today.
