Internal memos show the bill for dealing with child abuse in residential institutions has risen to €1.47bn, up from €1.36bn in 2011.
However, writing in today’s Irish Examiner, Mr Quinn has admitted that taxpayers are picking up the majority of the tab.
"I believe there is a moral obligation. I believe that fairness demands such an approach," he said.
"However, the response from the religious congregations remains disappointing.
"There has been no general acceptance by the congregations that they should meet a 50% share. Instead, the taxpayer continues to shoulder the brunt of these enormous costs."
And he asked the orders to quickly release the cash that has been promised for the Statutory Fund.
"I urge those congregations involved to commit the balance of the €110m already promised quickly so that we can continue to support survivors of institutional abuse," he said. "It is the right thing to do."
Documents obtained by the Irish Examiner show there is a growing shortfall in the contribution from the 18 congregations involved with the value of their offers continuing to erode.
It has left the exposure of the taxpayers to the redress scheme at approximately €1.1bn — 50% more than the target set by the Government at a meeting with the orders in Apr 2010.
After the Ryan report was published in 2009, the orders committed to increase their contribution to €476m. But this included €235m worth of properties, which the department has now valued below €60m.
Records of difficult negotiations over the last two years, released under the Freedom of Information Act, show the Government’s commitment to share the burden equally has been met with hostility from many orders.
One of the most powerful orders, the Sisters of Mercy, told Mr Quinn that the rising costs of redress were the State’s fault, and its responsibility.
Many orders considered the Government’s policy of pursuing a 50:50 contribution as a "bridge too far" and a policy that would bankrupt them.
Divisions have also opened up within the 18 orders about the portion of cost being carried by individual congregations.
The internal papers show at various meetings the orders told Mr Quinn:
* The 50:50 target was a flawed one which the orders had no legal obligation to respond to;
* Accepting a 50:50 principal would effectively suggest all members of orders were guilty of abuse;
* The cost of redress was not shared equally among the orders or other management bodies;
* The cost of the Commission to Inquire into Child Abuse should have been kept separate from the redress bill and not be borne by the orders
Cabinet records revealed the terms of its decision to refuse a number of different alternative offers from the orders. This was due to the complexity of the arrangement or that they conferred no benefit to the State.
Successive memos show the department originally expected to have all but €20m out of €107m in cash offers by 2014.
But it has now said it is unsure when €37m of the amount will be released by the orders.