Fr Brian D'Arcy, from Enniskillen, said he has had "countless conversations" with mothers faced with the difficult decision of having to travel to England for a termination.
Abortion is outlawed in Northern Ireland except in cases where the mother's health is at risk.
He told the Hot Press magazine: "I have sat with mothers, night after night after night.
"And I have always said to mothers, 'Whatever you choose is the right choice.'
"I always say that to mothers. My own view is that we should try to save all lives."
Fr Brian D'Arcy says he is pro-life but thinks it is important to help cover the costs in abortion cases where women travel for a termination, for example cases involving fatal foetal abnormalities.
He has proposed the money is used to bring the infant's remains back to Ireland to be buried.
"Whatever about the abortion, I have no problem at all having respect and love and care for the little infant whose life didn't get a chance," he said.
He went on: "So, whatever we should do to make the mother and father of that infant good, so much the better. I have no problem with that. In fact, I think it's a good thing to do be compassionate and loving in that situation."
He is speaking after a paediatric pathologist in Northern Ireland quit after claiming the region's strict abortion laws forced a couple to abort their baby in England and bring home the remains in a picnic cooler bag.
Outlining her reasons for resigning, Dr Caroline Gannon was particularly critical of the stance of Northern Ireland's attorney general in regard to abortion in the cases of fatal foetal abnormalities.
The couple travelled to England for the termination and had to transport the remains of their baby home for post-mortem in a cooler bag with ice packs.
Fr Brian D'Arcy finished saying he thinks it is inevitable that the Irish government will repeal the Eighth Amendment to the State's constitution, giving the unborn an equal right to life with mothers in all circumstances.
