Connolly, 68, an independent candidate backed by the entire left-dominated opposition, was likely to win more than double the votes of her nearest rival, an official from one of the governing parties said, citing an incomplete tallying of votes.
Fine Gael General Secretary John Carroll told national broadcaster RTE that the data suggested Connolly would win more than 60 per cent of the votes, with his party's candidate - ex-cabinet minister Heather Humphreys - probably finishing in the high 20s.
"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 second main governing party Fianna Fail, added.
A final result will likely be declared later on Saturday.
A long-time critic of the European Union in overwhelmingly pro-EU Ireland who has repeatedly denounced the EU's plans to boost military spending, Connolly was far from a household name at the start of the contest for the largely ceremonial role.
Many of her views - from questioning the trustworthiness of the United States, Britain and France over their stance on the war in Gaza to comparing German rearmament spending to that of the 1930s - are well to the left of many of the parties backing her, as well as outspoken incumbent Michael D Higgins.
Ireland's president is largely a figurehead, with seldom-used powers to test the constitutionality of legislation, but often speaks on the global stage and welcomes other heads of state to the country.
Connolly is a former clinical psychologist and barrister who served as deputy speaker of Ireland's lower house after being first elected in 2016.
She built momentum throughout the campaign by criticising government policy in areas such as housing, which found wide appeal among younger voters in particular.
She also benefited from dire campaigns from the recently re-elected centre-right coalition.
Humphreys was hastily selected after Fine Gael's first choice candidate withdrew due to illness.
Fianna Fail's pick, former Gaelic football coach Jim Gavin, abandoned his bid early on due to a financial scandal.
