The Housing Minister says he will strive for balanced regional development if elected as Fine Gael leader and Taoiseach.
He has held positions as Fine gael party leader ad ministerial portfolios in health, industry and commerce, and justice.
"While over the years we have had our differences on many political and policy issues, I have always appreciated Enda's decency, good humour and approachability", he said.
Mr Kenny stood down from the helm of Fine Gael at an emotionally-charged meeting of party colleagues in Leinster House in Dublin after delaying the announcement for several months.
Education minister Richard Bruton is also reported to be assessing the possibility of entering the race.
"Whatever I do next it will be different, not politics", he said.
Supporters of Varadkar last night claimed their man has the charisma to be the country's next leader - as Simon Coveney insisted: "People are not looking for a Taoiseach with an X factor".More news: 22 dead in terror attack at Ariana Grande concert in Manchester
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Leo Varadkar, 38, may appear an unlikely front-runner in what is still regarded as a socially conservative, relatively homogeneous country.
Mr Coveney dismissed the idea of cutting short the contest and said he has never lost an election.
Ms Doherty said she was supporting Mr Vardkar for his intellligence, integrity, sincerity and ability to listen.
Earlier on Friday the Waterford TD John Deasy, who is personally close to both candidates, came out in support of Mr Varadkar.
The contest is decided by an electoral college that gives the parliamentary party, made up of TDs, senators and MEPs, 65% of the vote.
From Thursday to Sunday next week, there will be four regional debates or hustings - in Dublin, Carlow, Ballinasloe and Cork.
Thanks to Enda Kenny, Fine Gael has an abundance of talented representatives, who have an ever increasing experience of governing.
The address will take place at lunchtime in the city centre where the Taoiseach will also meet members of the Global Irish Network and clients of Enterprise Ireland.