Leadership hopeful Varadkar, now Minister for Social Protection, visited Donegal last week.
But Kenny said he will remain on as Prime Minister until a new party leader is elected.
"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 two frontrunners to be the country's next PM are minister for social protection Leo Vradkar, who is gay and the son of an Indian immigrant, and Simon Coveney, who has served as the defence minister and whose father was also a government minister.
Ministers Charlie Flanagan and Paul Kehoe, as well as TD Fergus O'Dowd, have also backed Varadkar.
"I want to lead a party and a country that can heal numerous divisions that have opened up over the last decade or so".
Enda Kenny will step down as Fine Gael leader after nearly 15 years and not lead party into election. One that says to people that if you contribute to the system, the system is there to support you.
Both Coveney and Varadkar have served in Kenny's cabinet since 2011 and are unlikely to usher in major policy changes.
Man who ran over pedestrians was 'hearing voices'
The New York Police Department says the driver drove along the sidewalk for at least three blocks, hitting pedestrians as he went. A woman who used to live in the building, Fati Razak, said she occasionally sees Rojas when she returns to visit her mother.
"He also was part of reducing the Value-Added Tax rate for the hospitality sector, if you talk to any hotelier or anyone in the hospitality sector and they'll say that's one of the major reasons they can attract overseas visitors to Ireland".
BoyleSports can report support for Simon Coveney on Thursday morning and as a result, the Cork native is now 6/4 from 7/4 to be the new Fine Gael leader.
"His whole adult life has been devoted to public service and his personal integrity has never been questioned, even by his most ardent opponents", she said.
His supporters hope to narrow Mr Varadkar's lead, or pull ahead of him, before the series of hustings that will take place next week and mark the engagement between the candidates and rank-and-file members.
The 66-year-old had been under pressure to resign from factions within his own party that were dissatisfied with his leadership.
While Mr Varadkar is expected to personally launch his bid on Saturday, Mr Coveney fired the opening salvo on the steps of his party headquarters and declared he was not going to engage in sound bites and glib put-downs. He subsequently took up positions including minister for agriculture, defence and now housing.
Minister of State Michael Ring and Taoiseach Enda Kenny have yet to announce who they are backing in the contest.