Early tallies of votes after counting began at 0800 GMT on Saturday showed a wide lead for Connolly, 68, an independent candidate.
She is a long-time critic of the European Union in overwhelmingly pro-EU Ireland and was far from a household name at the outset.
"It does appear likely that Catherine Connolly will be elected and we now move on to the job of working with her as a government," Higher Education Minister James Lawless, a member of the Fianna Fail party, told national broadcaster RTE.
Ireland's president is largely a figurehead, with seldom-used powers to test the constitutionality of legislation.
Education Minister Helen McEntee also said it looked as though Connolly would win.
Her centre-right Fine Gael party's pick Heather Humphreys was the only other candidate left campaigning.
Fianna Fail's candidate abandoned his bid just days into the campaign.
Labour Party leader Ivana Bacik, part of the entire left-wing opposition bloc that backed Connolly, said it was "almost certain" that Connolly would be the 10th president of Ireland.
Connolly, a former clinical psychologist and barrister, was first elected to parliament in 2016, serving as a deputy speaker of Ireland's lower house.
In brief remarks to the media at her local County Galway count centre, Connolly thanked those who voted for her and said she would do her utmost to represent those who did not.
A final result will likely be declared later on Saturday.
