Abortion is only legal in the Republic of Ireland if the mother's life is at risk.
The Citizens' Assembly - made up of 99 members and a chairperson- voted in favour of the change on Saturday.
91 members took part in the secret ballot, RTÃ‰ reports.
44% voted to repeal or delete Article 40.3.3 of the basic law and 56% voted in favour of replacing or amending it.
It will now discuss the recommendations it will make to legislators in more detail.
A body set up advise the Irish government on constitutional change has voted to replace or amend the part of Ireland's Constitution which strictly limits the availability of abortion.