He will face housing minister Simon Coveney after nominations closed for the leadership of the governing centre-right Fine Gael party.
If Mr Varadkar wins the contest, he is set to become Ireland’s first openly gay Prime Minister.
The 38-year-old is favourite to take over from 66-year-old Mr Kenny, who is stepping down after six years as Taoiseach and 15 years at the helm of Fine Gael.
Mr Kenny’s successor is due to be elected by 2 June and Ireland’s parliament is set to vote in the new leader as PM a few days later.
Mr Varadkar has won early support for his leadership bid from several senior cabinet members and a majority of his parliamentary colleagues have publicly backed him.
The Dublin-born son of an Indian immigrant father and Irish mother, he became the first openly gay cabinet minister in Ireland after coming out in 2015.
He has campaigned on same-sex marriage and liberalising abortion laws, although like the majority of his party colleagues, he is an advocate of tight fiscal restraint.
He was a doctor before winning a seat in parliament in 2007 and has rapidly risen through the ranks, holding several ministerial portfolios.
In recent weeks, his campaign to clamp down on welfare cheats has been fiercely criticised in some quarters.
Despite picking up early backing in the race, he remains cautious, saying: “I’m not counting my chickens. I’m really humbled at the level of support I have received from my colleagues and I am really looking forward to the hustings and the debates.”
Ahead of the deadline for leadership nominations, Mr Varadkar had secured 45 of 71 available votes from members of Fine Gael’s parliamentary party.
They account for 65% of the electoral college, with 22,000 ordinary party members sharing 25%. County councillors account for the remaining 10%.
Cork-born Mr Coveney, 44, would need to win a significant majority of non-parliamentary votes to win, something that few commentators believe is likely.
He has been forced to deny that the contest was all but over.
“Leo’s got off to a good start but there’s two weeks to go so we’ll see how that plays out,” he said.