Education Minister Ruairi Quinn has said he believes that Cardinal Seán Brady should consider his position, following allegations in a BBC documentary about a church inquiry into child abuse in the 1970s.
Mr Quinn said this was appropriate because Cardinal Brady is the head of the Catholic Church in Ireland, which is the patron of 92% of the 3,200 primary schools.
He said that the Catholic Church also should consider the appropriateness of having at its head someone who had "failed spectacularly to protect children".
The documentary alleged that claims made by a boy in 1975 - to a church inquiry which Cardinal Brady was involved in - were not passed on to parents of other victims, or to gardaí or police.
Abuse survivors and support groups have also called on Cardinal Brady to resign.
Earlier, Tánaiste Eamon Gilmore told the Dáil that anybody who did not deal with knowledge of child abuse should not hold a position of authority.
Mr Gilmore described the revelations about further cases of child abuse at the hands of Brendan Smyth as "another horrific episode of failure by senior members of the Catholic Church to protect children".
He said the BBC programme misrepresented his role in the case, describing him as an investigator rather than a note-taker.
He also rejected any accusation that he had deliberately refused to take responsibility.
Abuse victims want Brady to resign
The abuse survivor at the centre of the controversy about the alleged cover-up has echoed calls for Cardinal Brady to resign.
Brendan Boland's made his remarks in a pre-recorded interview with RTÉ’s Prime Time, which will be broadcast tonight.
Another man who was abused by Brendan Smyth in the period after a church investigation in 1975 earlier called for Cardinal Brady to resign as Catholic Primate of All Ireland.
Sam Adair was interviewed by church officials, including the then Fr Brady, in 1975.
Mr Adair said that at the time of the investigation, Cardinal Brady was a skilled canon lawyer and not simply a note-taker as he described himself.
Mr Adair said that Cardinal Brady should step down and that the church should compensate Smyth's victims.
"This Titanic embarrassment, humiliation and injustice to Catholic children has never, ever, ever been dealt with by the cardinal whatsoever," Mr Adair said.
"Maybe someone else could step into the breach here and take this situation and solve this situation. He knew of five children's names and addresses and to have a rabid paedophile of the Catholic Church visiting those homes and sexually molesting those children, the very, very, very least he could have done was went and made sure that he slept at night with his clear conscience that the parents of these children knew that this tea-drinking, Marie biscuit-eating paedophile was not lurking around their houses. He did not keep these children from this devil in a dog’s collar."
Mr Adair said that times are different, but that the cardinal's own conscience should have been guideline enough to motivate him to ensure children were not further abused, saying that the cardinal had lacked "moral courage".
"Anybody walking through a park who saw some sort of instance of abuse, surely would have intervened or reported it to the police or made a phone call,” Mr Adair said.
"He done (sic) absolutely nothing and the most important part about this is that this isn't an ordinary man that's walking the streets - an ordinary 4x2. This is a man who is supposed to be the telephone line to God and people are relying on their eternal salvation on him. And this man hasn't even got the moral courage to report the child rape and the abuse of children."
A spokesperson for survivors of abuse in Northern Ireland has said "it is time" for Cardinal Brady "to go".
Margaret McGuckin of the group Survivors and Victims of Institutional Abuse has said it is a "disgrace" that Cardinal Brady has not resigned.
Speaking on RTÉ's Six One News, Ms McGuckin described as "shambolic" the stance Cardinal Brady has taken.
She urged him "to be humble and step down" before, she said, the Church in Ireland falls apart.
McGuinness calls for Brady to reflect
Northern Ireland's Deputy First Minister Martin McGuinness has said "speaking personally" he believes Cardinal Brady "should reflect on the wisdom" of his position that he will continue on as leader of Ireland's Catholics.
The Sinn Féin politician said: "I think that many Catholics, of which I am one, Catholic priests amongst whom I have many good friends and the public in general will be dismayed at these new allegations."
He described the situation as a "very grave" one for survivors of abuse, for the Catholic Church and for Catholics across Ireland.
Fianna Fáil leader Micheál Martin earlier said Cardinal Brady should consider his position given the enormity of the issues involved.
Independent Senator Rónán Mullen said it was not helpful for "politicians to be calling on church men to either go or stay".
He said he did not accept the argument that bishops' roles in school patronage gives politicians some kind of automatic right regarding church matters, "given that the church is probably ahead of the State now in child protection matters".
Mr Mullen said some of the comments by politicians today were an example of "bad governance".
SDLP leader Alasdair McDonnell said the leadership of the Catholic Church needed to think quickly and act decisively.