An exit poll for broadcaster RTE suggests the 77-year-old Higgins received 58 percent of the vote in an election contested by six candidates. A poll for the Irish Times newspaper suggests he secured more than 56 percent.
A vote share of more than 56.3 percent would be the highest ever achieved on a first ballot, beating the result won by longtime independence leader Eamon de Valera in a two-way contest in 1959.
Voters were also asked if they wanted to remove the offence of blasphemy from the Irish constitution. Exit polls suggest the measure will pass by a significant majority.
Results are expected later Saturday.