Counting of votes in the leadership contest is under way in the round room of the Mansion House in Dublin.
The winner will take over from Enda Kenny as taoiseach (prime minister) in the next few weeks.
The favourite is Leo Varadkar, the son of an Indian immigrant and Ireland's first openly-gay government minister.  The other contender is Simon Coveney.
Voting took place this week around the country.
It is decided in an electoral college system that gives 65% of the vote to the Fine Gael parliamentary party - made up of 73 TDs (members of the Irish assembly), senators and MEPs.
The party's 21,000 rank-and-file members have 25% of the vote, and 235 local representatives 10%.
Mr Varadkar took a commanding lead on the first day of the contest after winning the public support of many of Fine Gael's TDs (MPs).
Mr Coveney vowed to remain in the contest, and told RTE he remained optimistic about his chance in the leadership race.
He is 44 and Mr Varadkar is 38, so either of them would be Ireland's youngest ever prime minister.
Mr Varadkar, who came out as gay in the run up to the 2015 same-sex marriage referendum, is currently in charge of the welfare system.
He has come to personify the liberalisation of a country which was once regarded as one of Europe's most socially conservative nations - homosexuality was illegal until 1993.
However, Mr Varadkar has come under criticism for his comments on progressive issues and workers rights.
The new Fine Gael leader will have to be endorsed by the independent members of the minority coalition government before he becomes taoiseach. That's expected to happen later this month.
Enda Kenny, 66, who led the party for 15 years and was elected taoiseach in 2011, resigned as Fine Gael leader in May.
The biggest party in Ireland's ruling coalition, Fine Gael, will announce its new leader later.