Mrs Foster has called for closer ties between the Republic and the North
Fianna Fáil has called on the DUP leader Arlene Foster to 'follow her words with action' as we head towards Brexit.
Mrs Foster called for closer relations between the UK and Ireland in a speech in Killarney on Saturday.
She suggested that leaders on both sides of the border should work together for the benefit of everyone.
Mrs Foster said: "I often think that Northern Ireland and the Republic of Ireland are like a semi-detached house.
"The houses may look the same on the outside, but, inside, they look different and we do many things very differently.
"But no matter how contrasting the interiors are, they are tied together and part of the same neighbourhood and what happens on one side of the fence inevitably has an impact on the other.
"I know that we are rivals in some respects, but in so many ways success for one of us is success for the other.
"As we chart a new course for the future, it is not in our interests to see the Republic of Ireland do anything other than prosper."
"The Democratic Unionist Party supported the UK leaving the European Union but in so doing Brexit is not about pulling up the drawbridge, building a wall and cutting ourselves off from our nearest neighbours.
"But we must all recognise that change is coming as a result of the referendum.
"It is our job as politicians to help shape that change but to do so in a way that ensures that those economic, cultural and social ties that have endured through difficult times and have thrived through better ones continue into the future.
"We want to avoid a cliff edge for businesses by having a strictly time limited implementation period."
Sinn Féin's leader in the North, Michelle O'Neill, said the speech signalled a difference in tone but not in policy.
Fianna Fáil spokesperson on foreign affairs, Dara O'Brien, says words are good but they are not enough.
"What we really need to see is those words back up by actual action - the best way to do that is to get the (Stormont) assembly back up and running, to have an executive in the North of Ireland.
"We're 12 months without it and the DUP and Sinn Féin themselves share responsibility for the fact that the people in the North have no representation right the way thought this Brexit process.
"There's no question that we need closer cooperation between North and South, and east and west - indeed between Britain, particularly as we go through this Brexit process.
"Words are good, but they need to be followed up with action".
"I would ask Arlene Foster in a respectful way to follow through on her words in Killarney - which are welcome - but to follow through on the basis of extending the hand to Sinn Féin, making sure that both parties actually stand up to their responsibilities, which are representing the people in the North".