Murray is understood to have fought to keep his job during his discussions with officials at the Congregation for Bishops, with some sources suggesting he has become increasingly emboldened in his approach since arriving in Rome last week.
This is despite previous suggestions that he had travelled to Italy to tender his resignation last week.
However, it is believed that he has met with strong resistance amid a determination by Pope Benedict – whose reaction to the report last week has nevertheless been strongly criticised by abuse survivors – to draw a line under the affair.
Murray's situation is understood to have become virtually "untenable" following the intervention by the Catholic primate Cardinal Seán Brady last weekend, who said in an RTE interview that he was sure Murray "would do the right thing".
As a result, any decision by the Congregation for Bishops to suggest Murray should remain in situ would effectively be seen as an unprecedented challenge to his authority.
Similarly, a decision by Murray not to abide by the advice of the Congregation of Bishops that he should resign would be seen as a major act of disobedience to the Pope by a serving Bishop and as a result is thought to be extremely unlikely.
Speaking yesterday, Murray's spokesman said he was aware that there had been suggestions the bishop, who met with Cardinal Brady in Rome last week, has been fighting to save his job.
However, he claimed his position remained that if he felt his ministry was not the best thing for the Diocese of Limerick, "I wouldn't have to be forced."
"Bishop Murray travelled to Rome this week to meet with senior officials from the Congregation for Bishops," the spokesman said in an official statement.
"He remains in Rome as he is waiting to hear back from the Congregation with regards to a follow up meeting."
Murray's battle to keep his job comes as the Catholic church here braces itself for the possibility of an unprecedented series of mass retirements or resignations by current members of the Irish Bishops' Conference from next year, after Pope Benedict signalled that the hierarchy faces a major overhaul following the Murphy report's findings.
An editorial in the current edition of the Irish Catholic calls for the number of dioceses in Ireland to be halved from 26 to 13, and for greater transparency in the appointment of Bishops.
If implemented, such a move could see many of the current members of the Irish Bishops' Conference, a large number of whom are already in their 70s, either retiring earlier than planned or leaving their posts as bishops.
However, representatives of abuse survivors point to the fact that regardless of any future reorganisation of the church, several other former or serving auxiliary bishops in Dublin last week refused to resign on foot of the Murphy report.
This is despite suggestions emanating from the Vatican that Murray might be one of a number of bishops to leave their posts.
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