Polls by The Irish Times and RTE suggest about 69% voted to repeal a part of the constitution that effectively bans terminations.
Taoiseach (prime minister) Leo Varadkar, who supported the reforms, said it looked as if the country was about to "make history".
Official counting of votes will begin at 09:00 local time.
Those taking part in Friday's referendum were asked whether they wanted to repeal or retain a part of the constitution known as the Eighth Amendment, which says an unborn child has the same right to life as a pregnant woman.
Broadcaster RTE's exit poll suggested 69.4% in favour of the Yes side and 30.6% for No. In Dublin, 79% of people voted for repeal, according to the RTE poll.
An exit poll released by The Irish Times points to 68% Yes to 32% for No.
Tweeting on Friday night, Taoiseach Leo Varadkar said: "Thank you to everyone who voted today. Democracy in action. It's looking like we will make history tomorrow... "
Close result expected in Ireland's abortion referendum
In other reaction:
- Irish health minister Simon Harris tweeted: "Will sleep tonight in the hope of waking up to a country that is more compassionate, more caring and more respectful"
- Prominent No campaigner Cora Sherlock expressed disappointment at the exit polls but said the pro-life movement would "rise to any challenge it faces". She added: "Let's go into tomorrow with this in mind"
- Penny Mordaunt, the UK's minister for women and equalities, called it a "historic and great day for Ireland and a hopeful one for Northern Ireland". Northern Ireland's abortion laws are much stricter than the rest of the UK
- The leader of Northern Ireland's centrist Alliance Party, Naomi Long, said it appeared to be an "incredible result" for the Yes campaign
Currently, abortion is only allowed when a woman's life is at risk, but not in cases of rape, incest or fatal foetal abnormality.
The turnout looks to have been higher than that for the country's referendum on same-sex marriage and its most recent general election.
More than 3.2 million people were registered to vote in the referendum, with more than 100,000 new voters registering ahead of the poll.
The referendum was the result of a decades-long debate about abortion in the Republic of Ireland and was the country's sixth vote on the issue.
Where does the law stand?
The now-controversial Eighth Amendment was introduced after a referendum in 1983.
It "acknowledges the right to life of the unborn and, with due regard to the equal right to life of the mother, guarantees in its laws to respect, and, as far as practicable, by its laws to defend and vindicate that right" - meaning the life of the woman and the unborn are seen as equal.
Since 2013, terminations have only been allowed in Ireland when the life of the mother is at risk, including from suicide.
The maximum penalty for accessing an illegal abortion is 14 years in prison.
In 2017, the Citizens' Assembly, a body set up advise the Irish government on constitutional change, voted to replace or amend the part of Ireland's Constitution which strictly limits the availability of abortion.
So the Irish people were asked if they wanted to remove the Eighth Amendment and allow politicians to set the country's abortion laws in the future.
The wording on the ballot paper was: "Provision may be made by law for the regulation of termination of pregnancies."
The ballot paper did not mention the Eighth Amendment or abortion, instead asking: "Do you approve of the proposal to amend the Constitution contained in the undermentioned Bill?"
Those who wanted to retain the Eighth Amendment voted No, while those who wanted to replace it voted Yes.
If a majority has voted yes - as appears to be the case - then the Irish government's recommendation is that women will be able to access a termination within the first 12 weeks of their pregnancy.
However, beyond 12 weeks, abortions would only be permitted where there is a risk to a woman's life or of serious harm to the physical or mental health of a woman, up until the 24th week of pregnancy.
Terminations would also be permitted in cases of fatal foetal abnormality.
The official result is expected early on Saturday evening.
