The 38-year-old will face Housing Minister Simon Coveney after nominations closed for the leadership of the governing centre-right Fine Gael party, reported Sky News on Saturday.
Varadkar is favoured to take over from 66-year-old Kenny, who is stepping down after six years as Taoiseach and 15 years at the helm of Fine Gael.
Kenny's successor is due to be elected in the June 2 general elections and Ireland's Parliament will vote for Prime Minister a few days later.
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, Leo Varadkar became the first openly gay cabinet minister in Ireland after coming out in 2015, reports Sky News.
He has campaigned on same-sex marriage and liberalising abortion laws and 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.
Ahead of the deadline for leadership nominations, Varadkar had secured 45 of 71 available votes from members of Fine Gael's parliamentary party.
They account for 65 per cent of the electoral college, with 22,000 ordinary party members sharing 25 per cent. County councillors account for the remaining 10 per cent.
(This story has not been edited by Social News XYZ staff and is auto-generated from a syndicated feed.)
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