The Association of Catholic Priests (ACP), which has more than 1,000 members, has published correspondence it sent to the cardinal seeking a meeting with bishops to discuss the future of the church.
In response, the Bishops' Conference executive secretary Fr G Dullea said the hierarchy had decided that engagement with the ACP should happen at local level.
The council is organised on diocesan level and involves priests meeting several times a year to discuss issues and then report to their local bishop. It does not allow for the ACP as a group to meet face-to-face with bishops.
It is the latest snub to the ACP after bishops in the west of Ireland refused to attend an assembly of priests and lay people in Galway last month and signals that relations between the hierarchy and the ACP are at a new low.
Fr Tony Flannery, a spokesman for the ACP, said they were disappointed and saddened with the response to their letter. He added that it was a "fairly definitive statement" on the hierarchy's feelings about the group.
Fr Flannery is a Redemptorist priest and was a long-time columnist with the order's 'Reality' magazine until earlier this year when he was silenced by the Vatican because of his liberal views.
He said: "Essentially what we have been told is for us to enter into discussion with ourselves. There is no episcopal involvement with them (the Council of Priests) at all. It's a total opting out of any discussion or dialogue with us which is very sad. There is nothing good about this and we don't take any pleasure in this."
Fr Flannery said that there had only been minimal engagement by the bishops with the ACP.
Survive
A delegation met with an episcopal commission to discuss the introduction of the New Missal last year, while earlier this year there was a private meeting between three members of the ACP leadership and two representatives of the bishops' conference.
In June, the ACP wrote to Cardinal Brady to request a meeting with the Episcopal Conference.
"We stress that the Association of Catholic Priests is not 'against' the church. We are part of it, we care about it and we want it to survive," they wrote.
A spokesman for the Bishops' Conference said yesterday there were "ongoing meetings" between bishops and priests at local level.