First Minister of Scotland Nicola Surgeon described Mr Kenny as a "good friend to Scotland".
Many consider ministers Leo Varadkar and Simon Coveney to be the favourites to fill the vacancy.
In a statement to colleagues in Dublin, Mr Kenny said he was retiring from midnight with his successor to be in place on June 2.
A number of Ministers have declared for the Leo Varadkar this morning - including Paschal Donohoe, Charlie Flanagan and Paul Kehoe.
DUBLIN, May 18 (Reuters) - Irish Finance Minister Michael Noonan said on Thursday he will step down when a replacement for Prime Minister Enda Kenny is chosen next month, having presided for six years over the dramatic recovery of the Irish economy and banking system.
He told his party he would remain as leader for a "brief but appropriate period" to facilitate a leadership contest.
"I will continue to carry out my duties as party leader in an acting capacity, until my successor is elected through the Fine Gael leadership election process", he said.
Ms Doherty said she was supporting Mr Vardkar for his intellligence, integrity, sincerity and ability to listen.
Cyber attack is biggest of its kind ever launched, say security chiefs
There is still no evidence any of our hospital systems were affected and we are completely confident no patient data was involved. Carmaker Renault said one of its French plants, which employs 3,500 people, wasn't reopening Monday as a "preventative step".
Earlier on Friday the Waterford TD John Deasy, who is personally close to both candidates, came out in support of Mr Varadkar.
Mr Varadkar has already secured the public endorsements of around 35 members of the parliamentary party.
Minister Dara Murphy suggested that Simon Coveney has the qualities needed to lead Ireland.
The parliamentary party holds the whip hand with 65 per cent of the voting strength, with councillors allocated 10 per cent and rank and file members 25 per cent.
"He understands, as he said last week, the unique circumstances of Donegal, cut off from the rest of the North for most of the last century by partition, and cut off geographically and in other ways from the rest of the State".
Irish Prime Minister Enda Kenny conceded defeat on 27 February following elections that saw the governing coalition punished by voters tired of austerity, leaving the eurozone country in political limbo with no clear victor.
The current housing minister has also been in charge of several portfolios including agriculture and defence.
Richard Bruton and Simon Harris have ruled themselves out of the running, as has Frances Fitzgerald - meaning the contest is now a two horse race.