Archbishop Diarmuid Martin has said that at this moment, he believes that decisions on whether to resign by Cardinal Desmond Connell and others mentioned in the Murphy Report should be personal.
This is the first time in the three-day controversy over his Church's protection of priests that he has hinted he may change his public position.
Earlier, Bishop Donal Murray of Limerick said his decision on whether to resign hinges on the views of his clergy and parishioners and on whether his presence in the diocese is a help or a hindrance.
The Commission of Investigation report into clerical abuse in the Archdiocese of Dublin described Dr Murray's response to an allegation of abuse as inexcusable, and criticised his handling of other allegations.
Dr Murray said that at no time did he receive an allegation of sexual abuse and fail to take it seriously or attempt to cover it up.
He said he could honestly say that in one case his inability to get to the full truth was due to a lack of skill and experience and was not the result of any lack of effort on his part.
He said this was no consolation to the children who were abused and that he would remain eternally sorry and apologetic to anyone whose suffering he might have prevented.
The founder of One in Four, Colm O'Gorman, has described as disingenuous remarks made by Dr Martin that, in the past, bishops who have stood down made that decision themselves.
Mr O'Gorman said the Vatican requested the resignation of former Bishop of Ferns Brendan Comiskey.
It did so under a code of Canon Law that is brought into effect when it is deemed that a bishop is not fit to carry out his function as bishop.
Mr O'Gorman said Dr Martin's failure to put the decision making, in the cases of those bishops criticised in the Murphy report, in the hands of the Pope, is disingenuous.
The Minister for Defence and Limerick TD, Willie O'Dea, has said that he was bitterly disappointed to read that the Catholic bishop of Limerick had behaved ‘inexcusably’ according to the Murphy Report.
Speaking on RTE's The Week in Politics, to be broadcast later tonight, Mr O'Dea said that Dr Donal Murray is a decent man who would think deeply about things.
The Minister said that he is sure that the Bishop would at this moment be examining his position and would make the ‘appropriate decision.’
On the same programme, the Fine Gael leader said those who knowingly moved priests, from parish to parish, who they knew had abused children should resign.
Mr Kenny declined to name any individual bishop saying it was not his job to stand in judgement of any man.
Meanwhile, the Bishop of Dromore Dr John McAreavey, speaking on BBC Radio Ulster's Sunday Sequence programme, said he would step down if he lacked credibilty.
Statements read at Masses across Ireland
Many of the country's other bishops have today made statements on the Murphy report to their congregations.
The Bishop of Ardagh and Clonmacnoise has described as 'revolting and shameful' the manner in which Church authorities dealt with allegations of child sexual abuse.
Bishop Colm O'Reilly, in a statement read at all Masses in the diocese, said the church had failed to act in an appropriate manner.
He said what made clerical abuse most abhorrent is that it was perpetrated by people with a sacred calling who betrayed the trust placed in them.
In his statement, he said the Church must accept that its leaders put the good of the institution before the welfare of the abused.
Bishop O'Reilly said a great wrong had been done and he urged anybody who had suffered at the hands of Church personnel to approach civil authorities or the diocese.
A pastoral reflection was also circulated at Masses this morning in the diocese of Down and Connor expressing horror and distress at the findings of the report.
Bishop Noel Treanor and his assistant bishop and priests said the heinous crimes against children described in the report were appalling and distressing in the extreme.