Speaking for the first time since the contest was announced, Mr Varadkar said he was not "counting his chickens yet". He became the longest-serving Fine Gael Taoiseach in April 2017.
Minister for Social Protection, Leo Varadkar and Minister for Housing, Simon Coveney are the clear frontrunners in the race.
Mr Donohoe stated his intention to vote for Mr Varadkar in an interview on RTÉ's Morning Ireland.
Mr Kenny delayed the transition to a new leader on a number of occasions this year, asking colleagues for time to visit the U.S. for the annual St Patrick's Day celebrations and to meet President Donald Trump, continuing the unique tradition Ireland has of access to the White House every March 17.
He made his decision public at lunchtime today along with a host of other ministers, including Richard Bruton. "That is precisely what we need right now and I believe Simon Coveney will deliver on this", Mr Harris said.
Mr Donohoe declared for his Dublin colleague while refusing to be drawn on what plum job he is hoping for in the next cabinet.
"His dedication and determination gave Fine Gael a landslide in that year's general election", he said.More news: Chelsea enjoy double title delight as Under-18s crowned Premier League champions
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He added "Simon Coveney has also seen his priced halved into 3/1 from 6/1 to be Taoiseach after the Next General Election and judging by the momentum behind the Cork native over the past 24 hours, he certainly can not be ruled out".
The 66-year-old had been under pressure to resign from factions within his own party that were dissatisfied with his leadership.
Roscommon Senator Frank Feighan said he would be supporting Minister Varadkar.
Mr Varadkar sought to set a strong early pace by organising a large number of declarations for his candidacy.
However, those campaigning for Mr Coveney, who has around 20 supporters so far in the parliamentary party, believe another 20 are undecided.
The regional debates, or hustings, will take place in Ballinasloe, Dublin, Carlow and Cork over four days next week.
The electoral college of the Fine Gael parliamentary party, county councillors, and party members will prepare to choose a new leader to be announced on 2 June.