At present the Catholic bishops are patrons of around 3,000 of the country's 3,200 primary schools.
Some of them were criticised in last week's Murphy report for not doing enough in relation to allegations of abuse when they were auxiliary bishops in the Dublin diocese.
One of them, Bishop of Limerick Dr Donal Murray, has come under strong pressure to step down.
The revelations sparked questions about the continued role of some of the bishops in the running of schools.
As patrons the bishops appoint chairpersons of boards of management and two members of the board of each Catholic school.
They also set up the assessment panels for appointing principals. The formal appointment of principals and teachers is sanctioned by the patron who also has responsibility for the ethos of the school.
The National Parents Council (Primary) wants the child protection services of the Health Services Executive to "immediately review the register of school patrons".
It said it was time the State took responsibility for child protection in Ireland.
"The decision regarding individuals remaining in certain positions or offices must have child protection as the primary concern," said the council.
"Responsibility must remain with the Government regarding the appropriate standards of governance in the Irish education system," it said.
Appropriateness
Taoiseach Brian Cowen has stated that it was for institutions and their members to determine the appropriateness of any individual to hold ecclesiastical office.
"However, in the Irish school system there is a direct relationship between an individual holding such ecclesiastical offices and other areas of responsibility, such as that of school patron," the council said.
"Parents and children need to be assured that any individual about whom there is a serious concern. . . regarding child protection will not continue to have influence and decision making responsibilities in our schools today," it added in a statement.
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