The call from the advocacy group, Irish Survivors of Child Abuse, says the Catholic Church religious orders which are signatories to the 2002 Indemnity Agreement with the State have reneged on their promises to pay their fair share towards the State's redress process.
Irish SOCA made its plea for Pope Francis to break the impasse concerning abuse in a statement this afternoon.
"Enda Kenny should travel to Rome as soon as practical and demand a comprehensive and honourable settlement of all matters connected with the child abuse scandals which implicate the servants of the Roman Catholic Church in Ireland.
"These matters have dragged on for too long."Owners of the largest Catholic institutions where children were abused while in State care have defended their contributions to the €1.5bn bill for paying redress to survivors.
The Christian Brothers and Sisters of Mercy were responding to: yesterday's report by the State's auditor that the 18 Catholic entities concerned had paid about 13% of the cost; and to criticism by Minister for Education Richard Bruton.
The minister had stated that the Church's progress towards shouldering its promised one-quarter share of the redress bill had gone into reverse.
Mr Bruton said he will continue to exert "moral pressure" on the religious orders for more progress in meeting their commitments to pay for the cost of residential abuse.
Speaking on RTÉ's Morning Ireland, Mr Bruton said he believed the Church should meet its responsibilities for endemic abuse where protection of the institution was prioritised over the protection of children.
In a statement the Christian Brothers' leader, Brother Edmund Garvey, said the audit report's 14-month-old figures do not take account of the congregations' more recent €14m cash payment.
Brother Garvey also highlighted that playing fields worth "well over €100m" are almost ready to be transferred to the Edmund Rice Schools Trust.
He calculated that the Brothers' total contribution to help their fellow-brothers' abuse victims would rise to over €600m.
Mr Bruton said the offer of transferring playing fields into the Edmund Rice Trust by the Christian Brothers was not acceptable - rather it should be transferred to the State or sold under a joint agreement.
In a separate statement, the Sisters of Mercy said they had honoured all of their commitments.
They recalled their 2009 commitment to make a payment valued at almost €128m, €81m of which was to be paid directly to the State in the form of properties, which were worth that sum at the time.
However, they said the financial downturn had eroded the State's actual financial gains from the property transfers.
The Sisters said that they had "always made clear that the value of (their) contribution was subject to the fluctuations in value attaching to individual properties".
Minister Bruton said the Government remains of the belief that an equitable share-out of costs would be 50-50 between State and Church, but we are "a far cry" from that.