Archbishop Eamon Martin (above) met campaigners in Armagh on Tuesday.
The Historical Institutional Abuse (HIA) inquiry is due to report to the Executive next January and could recommend a pay-out for survivors.
Margaret McGuckin from Survivors and Victims of Institutional Abuse said: "It is the responsibility of the Northern Ireland Executive to set up a redress scheme for victims of institutional abuse here, but the Catholic Church and religious orders which ran homes where we were abused have a moral and financial responsibility to victims.
"We have asked the Archbishop to tell Executive ministers that the Church stands ready to contribute significant funds to a government-run redress scheme.
"Other organisations, private or public, which ran homes where abuse took place bear a similar responsibility."
The Sisters of Nazareth congregation is among religious orders which have already admitted physical abuse and apologised.
