In a phone call with British Prime Minster Theresa May, Kenny warned that efforts to shore up her minority government with the hardline Protestant DUP should not put at risk the Good Friday peace accords.
"They discussed the outcome of the UK general election and the (British) prime minister outlined the proposed supply and confidence arrangement between her party and the DUP," a spokesman for the Irish government said.
Before the 1998 peace deal, Northern Ireland was plagued by violence over Britain's control of the province.
The Good Friday accords led to a power-sharing arrangement between nationalist, largely Catholic parties who want to join Ireland and the pro-British Protestant community.
Kenny also "noted the absence of any nationalist voice in Westminster", after Northern Ireland's Social Democratic and Labour Party (SDLP) lost its three seats in Thursday's election.
Sinn Fein won seven seats but the party refuses to sit in the House of Commons, which requires swearing allegiance to Queen Elizabeth II.
Kenny is due to hand over power this week to his successor as leader of the ruling Fine Gael party, Leo Varadkar.
"The taoiseach stated that there should be an early meeting between the prime minister and his successor Leo Varadkar and wished her well in the challenges that now lie ahead," the spokesman added.
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