The party will nominate the winner in the Irish Parliament to succeed Enda Kenny as Taoiseach.
With backing from most of the parliamentary party, welfare minister Leo Varadkar has a clear lead.
The 38-year-old would be Ireland’s first gay Prime Minister and the first of ethnic minority heritage.
“I come from a mixed background,” he explained. “My mother is Irish, my dad is Indian, although they met in England as doctors and nurses in Leicester a long time ago.
“Also, we had our marriage equality referendum in Ireland two years ago… so that social transformation has already happened and that has allowed somebody with my background to compete for this office.”
The other contender, housing minister Simon Coveney, believes he has most grassroots support.
He acknowledged his rival’s lead but refused to accept that the contest has already been won.
“This election is not over.
“We’ve heard from some politicians who’ve declared publicly who they’re supporting… but nobody has voted yet.
“What I’m about is talking to people about the future of this political party, where it wants to take Ireland as a country in the future.”
It is the first time the largest party in Ireland has chosen a leader using an Electoral College system.
Some 21,000 Fine Gael members and 235 local elected representatives are entitled to vote.
The leadership contest is the largest single party election since the foundation of the Republic.
But Ireland has seen such social change in recent years, the prospect of a gay premier is not even a talking point.
Lise Hand, political journalist, explained: “I think Irish people have processed it, if there is anything to process, and they are just concentrating on the two personalities… and the various policies.
“I think it is quite interesting that all the palaver about his sexual orientation is really just not pertinent to this particular race.”
Polling will continue for four days and the result is expected to be announced in Dublin on Friday.