An Irish Times poll of 4,000 said the "Yes" camp was leading by 68 percent to 32 percent. Another poll by national broadcaster RTE suggested an even bigger victory, with 69 percent to 30 percent backing reforms.
The abortion campaign had gathered steam after an Indian-origin woman died in 2012 after her requests to terminate a 17-week pregnancy were denied by the Catholic country. Savita Halappanavar has now become the face of the campaign in Ireland.
Ms Halappanavar, a 31-year-old Indian-origin dentist, had sought a pregnancy termination when told she was miscarrying, but the request was turned down due to Ireland's then strict anti-abortion laws. She died of blood poisoning days after miscarrying in October 2012.
The Irish Times survey suggested that women voted by 70 percent in favour of the proposal and 30 percent against. Support among men was 65 percent pro-choice and 35 percent anti-abortion.
"Thank you to everyone who voted today. Democracy in action. It's looking like we will make history tomorrow," Prime Minister Leo Varadkar, who backs the reform, said in a tweet.
People over 65, however, voted mostly against overhauling the current legislation, which only allows terminations in cases where the mother's life is in danger.
Among the youngest voters, 18-24-year-olds, the poll found that 87 percent of respondents voted to allow abortion.
The exit poll showed "an overwhelming desire for change that nobody has foreseen," wrote Irish Times deputy political editor Fiach Kelly.
"The victory for the 'Yes' campaign looks set to be neither narrow nor based on a few segments of Irish society. Rather, it will be carried high on the shoulders of a majority across the entire country," he said.
Kelly pointed out that in Connacht/Ulster, traditionally seen as the most conservative region in the country, the exit poll indicated that 59 percent voted for change.
He called it "the final casting off of old mores".
Nearly 3.5 million voters were asked whether they wanted to overturn the ban after an emotional and divisive campaign.
Results for different parts of the country are expected from 1100 GMT, with a final result to be announced later in the day at the main counting centre in Dublin Castle.
People arriving at polling stations from the early morning spoke about the momentousness of a complex decision that in many cases pitted moral against religious beliefs.
Ireland has traditionally been one of the most religious countries in Europe. However, the Roman Catholic Church's influence has waned in recent years following a series of child sex abuse scandals.
"The fact that it's illegal for somebody in Ireland to seek medical treatment, having to travel outside the country and to feel that guilt, shame and isolation, it's absolutely shocking," said Belinda Nugent, 43, a community activist voting in north Dublin.
"I took it really personally, this vote."
But across the city, Finbar O'Regan, 50, said he wanted a "good, strong No vote".
The unemployed Dublin voter said his mother had been sent to England to have him born and adopted.
"I'm a staunch No. It's the life of an unborn baby. I'm one of the lucky ones," he said.
At present Ireland's eighth amendment -- which could be repealed in the vote -- recognises the "right to life of the unborn" with an "equal right to life of the mother".
Canvassers were out on the city's streets holding up signs and wearing t-shirts with campaign slogans. One in southwest Dublin offered hugs to people who voted in favour of repealing the ban.
The eighth amendment to the Irish constitution was installed following a 1983 referendum which approved outlawing abortion.
Anyone terminating a pregnancy in Ireland could face 14 years in jail.
The ban has led to thousands of women travelling each year to neighbouring Britain, where terminations are legal, or increasingly turning to abortion pills sold online.
Since 1983, around 170,000 Irish women have gone abroad for terminations.
The law was tweaked in 2013 to allow terminations if the mother's life is at risk.
The Irish government has proposed that if the eighth amendment is repealed, abortion will be allowed up to 12 weeks and between 12 and 24 weeks in exceptional circumstances.