Fine Gael's executive council will meet to officially open the contest to become party leader.
The decision comes following a disappointing General Election result previous year, which saw Kenny's party fail to secure a majority in the Irish Parliament and criticisms about the way in which he has handled a police scandal. "Clearly, there is a lot of momentum for Leo today, and good luck to him".
Fianna Fail leader Micheal Martin said Kenny had worked diligently for his country over many years.
He said Mr Varadkar is "the most capable politician of his generation".
"They've got an early head start, there's no doubt about that", he told Clare FM. Some of these have yet to publicly declare their hand, however.
He said that Minister Varadkar has a "vision of a resilient and competitive economy" that can support a fairer Ireland.
The Social Protection Minister had been keeping tight-lipped on his plans but revealed he is "humbled" by the support he has received in the campaign so far.
Once the winning candidate has been safely ratified, party bosses are expected to switch their focus to planning for the next general election. He described it as the most hard political decision of his life.
The small opposition Greens called for the new prime minister to be in place by the June 22 gathering of European Union leaders.
Minister Flanagan joins Peter Burke as local supporters of Varadkar, while Senator Gabrielle McFadden and junior minister Marcella Corcoran Kennedy are backing Simon Coveney. "But I know that the party, and the country, will always remember him and will always be grateful".
"He would be a quite capable of leading the party and the other man Leo Varadkar would as well.but I would be sticking my neck out and saying I would be supporting Coveney", he said.
The contest will be decided using a weighted voting system across three electoral colleges - the parliamentary party, councillors and party members.
Yesterday, Enda Kenny announced that he would retire from the role of leader; a decision which came into effect at midnight on May 17. These include Tánaiste Frances Fitzgerald and Minister for Agriculture Michael Creed.