The two favourites are Leo Varadkar, who is the son of an Indian immigrant, and Simon Coveney, whose father was also a government minister.
Kenny, dismissed as a political lightweight before leading his party to its best ever election result in 2011, built up a close rapport with fellow European Union leaders alongside Finance Minister Michael Noonan, who a number of Fine Gael lawmakers expect will also stand down following the leadership contest.
In another significant development that further signalled generational change within Fine Gael, Minister for Finance Michael Noonan announced he would not seek a place in the next cabinet and would not stand at the next election. Since then his Fine Gael-lead coalitions have survived years of political turbulence and mass protests against austerity.
"There are two strong candidates in the race, and I know that neither will let the competition become divisive for the party or the country".
The Leader has since learned organisers have provisionally set their sights on June 9, the week after a new Fine Gael leader is expected to be ushered in.
He has asked that this be concluded by close of business on Friday 2 June.More news: Metro North Train Off Rails in Rye, No Serious Injuries
Spokespeople for Tanaiste Frances Fitzgerald and Education Minister Richard Bruton last night declined to say if they would be joining the race.
Fine Gael lawmakers and councillors make up 75 percent of the vote, with ordinary members accounting for the balance.
Kenny will remain as a party leader in acting capacity until his successor is elected.
"Enda gave hope to a battered and bruised Ireland in 2011", he said in tribute to Kenny.
Fine Gael party chairman Martin Heydon described Kenny as a "compassionate, pragmatic and inclusive" leader.
Kenny led Ireland through a difficult recovery from the 2008 global financial crisis, which hit the debt-fueled "Celtic Tiger" economy particularly hard. He said he contacted Mr Coveney yesterday morning to inform him of his decision but was not able to reach Mr Varadkar. While you're here.Check out Episode 3 of The JOE Show - Ireland's first social chat show - featuring live studio guests The Strypes, music from Jamie Duff, the best new comedy sketches and Ireland's biggest U2 fan goes in search of Bono and the boys. Mr Coveney will travel to Ennis today for a series of engagements, including the Clare Fine Gael AGM.