The Democratic Unionist Party (DUP) said the Taoiseach should "focus on his own problems" after his comments on a possible border poll.
On Monday Mr Kenny said the prospect of a future border poll on the re-unification of Ireland should be included as part of the negotiations between the EU and the United Kingdom on the Britain's departure from the bloc.
He said that the negotiations “should take into account the possibility, however far out it might be, that the clause in the Good Friday Agreement might be triggered and that if there is a clear evidence of a majority of people wishing to leave the United Kingdom and join the Republic that that should be catered for in the discussions that take place.”
DUP MP Gregory Campbell said on Tuesday that now was the "time for calm" and urged Mr Kenny to "focus on his own problems".
Mr Campbell said: "We have no interest in becoming embroiled in internal Irish politics, as Fine Gael try to respond to Fianna Fail.
"More heat than light at moment will not help the situation. We need to move forward to get the best deal for Northern Ireland and that includes talking to both Brussels and Westminster, and to the Irish Government where the cross-border element is relevant.
“The UK government has made its position clear on a border poll — consistent with previous agreements. The new prime minister noted in her speech outside 10 Downing Street that she believed in the Union and now is the time for all to work constructively to get the best solution.”
DUP MP Ian Paisley described the remarks by Mr Kenny as “pathetic and deliberately mischievous”.
Ulster Unionist Party (UUP) MLA Jenny Palmer described the idea from the Taoiseach that Brexit talks should take into account the possibility of a Border poll, as "much ado about nothing".
Mrs Palmer said: “Everyone who approaches the border poll issue with an ounce of sense recognises, as the Taoiseach did, that there has to be evidence that a majority is likely to vote to change our constitutional position before a border poll can be triggered. The bottom line is that the evidence simply does not exist.”
UUP MLA Doug Beattie tweeted:“Not in my lifetime, my children’s or my children’s children. One island. Two countries” .
Traditional Unionist Voice (TUV) councillor Timothy Gaston said Mr Kenny "should mind his own business" and described the prospect of a border poll resulting in a united Ireland as "fantasy".
Mr Gaston said: “As Kenny himself notes any such poll would require evidence that a majority of people in Northern Ireland wished to have a United Ireland. The evidence suggests the opposite.
“These comments have more to do with Nationalist politics than political reality.
Social Democratic and Labour Party (SDLP) leader Colum Eastwood said "the reunification of Ireland is the biggest and the best idea around".
Since the Brexit result Sinn Féin has repeatedly said it will push for a border poll as the people of the North do not want to be “dragged out of Europe”.
Sinn Féin president Gerry Adams said "after some initial criticism of our proposal the Fianna Fáil leader and the Taoiseach have come around to this position also".
People Before Profit MLA Gerry Carroll said his party “stands for a 32 county socialist Ireland”.
“In the aftermath of Brexit, we are for the right of the people of the North, and Ireland as a whole, to hold referenda on whether or not the border should remain,” he said.
“On Monday the new Secretary of State for Northern Ireland James Brokenshire rejected calls for a border poll during his first visit to the North.
