The couple - who have not been named, revealed on BBC Radio Ulster's Nolan Show on Tuesday that Fr Damien Quigley contacted the couple via private message.
A listener later contacted the show to explain what the message said.
It is alleged Fr Quigley wrote to the couple and said: "Just a heads up, that as the priest you've asked to celebrate your wedding for you, you and I would need to have a conversation about any promotion you may do to advocate for abortion in Ireland.
"Such promotion or advocacy would impact on the appropriateness of me celebrating your wedding or possibly the wedding taking place in a Catholic church.
"I'm sure we can have a good chat over a coffee sometime."
Fr Quigley confirmed the message to host Stephen Nolan but said he wouldn't comment further as the matter is private.
However, he did issue a statement, which said: "From the outset, I wish to advise you that it would not be appropriate for me to discuss details of my pastoral support to any specific individual or couple in their preparation for the sacrament of marriage," the statement read.
"However, please be advised that I have never refused to prepare any person or couple for the sacrament."
The Catholic Church also distanced themselves from the priest's message.
A statement released by the Church said: ""It would not be appropriate to discuss details of a priest's pastoral support to any specific individual or couple in their preparation for the Sacrament of Marriage.
"However, please be advised that Father Quigley has never refused to prepare any person or couple for the Sacrament."
Stay up to date with the latest news stories from a Christian perspective. Sign up to our daily newsletter and receive more stories like this straight to your inbox every morning.
