Kenny wants second term in power for FG and Labour
But there is no sign of the government parties having a formal vote transfer pact in the local and European elections.
Speaking at the Fine Gael European election selection convention in the Ireland South constituency, Mr Kenny said there were only 100 weeks to go to the next general election.
"I'd like to think that when we fulfil our mandate at the end of two years (from now) that the people will be prepared to put in this Government, the make-up of this Government, back for a second term." Mr Kenny said this appeal was not a sign of a transfer pact emerging between the two parties.
"No, not really. We'll consider that later on. But obviously both parties are doing their own selections and their own candidates and getting on with that," he said.
Mr Kenny reiterated the party was looking to win five seats in the European elections – one in Dublin, two in Ireland Midlands-North-West and two in Ireland South.
In Ireland South, which covers Munster and south Leinster, FG selected two candidates yesterday – sitting MEP Sean Kelly and Senator Deirdre Clune. The Taoiseach said the party would add a third candidate from the south-east in due course.
The delay in finalising the ticket follows the decision of former IFA president John Bryan not to run for the party.
FG sources say the candidate will be selected in the next fortnight, with polling being conducted to assess the prospects of party TDs Simon Harris and Andrew Doyle, both from Wicklow, and Senator Michael D'Arcy from Wexford.
Agriculture Minister Simon Coveney said the party could take two seats out of four in the constituency.
"Anything else will be a failure for this party," he said.
Ms Clune, the daughter of former Foreign Affairs Minister Peter Barry, pointed to her family's tradition in the party. "I was born with a Fine Gael membership card in my hand. And my house was steeped in Fine Gael," she said.
She added that she was "not just a name on a ballot paper".
"Deirdre Clune is a serious candidate that will fight tooth and nail to win a seat," she said.
Mr Kelly pointed to his record as an MEP for the past five years. He said he had the best record among Irish MEPs for voting in the European Parliament, only missing two votes.
He said he had given 726 speeches – twice as many as his nearest rival: "We need people with experience to have Ireland continue at the heart of Europe."
In his address to the convention, Mr Kenny said the Government would be judged on whether it has sorted out public finances and whether people had an opportunity to get jobs.
He praised the passing of the Haddington Road Agreement with the public sector, which he said was the only deal of its type in Europe, and paid tribute to the role of Public Spending Minister Brendan Howlin.
Mr Kenny said he hoped FG and Labour would get a second term in office together.
"And I say that because I know this country and I really do feel that a second opportunity allows you to put in place the base and the foundations and structure for the generations that are to come."