The report contains the attitudes of parents of 2,544 primary pupils and 915 pre-school children, and recommends school buildings be freed up to allow an Educate Together school be set up in each of the five areas.
Although fewer than 25% of parents with children aged 12 or under responded in two areas, support for a wider choice of patronage ranged from 37% in Castlebar, Co Mayo, to just over half in Tramore, Co Waterford.
The numbers who would send their children to a school run by an alternative patron was between one quarter and just over one-in-three, and 35% to 44% did not support a wider choice of patrons.
The five areas are among 44 identified as having little or no choice of primary school other than those under the patronage of local bishops.
Among parents who want greater choice, multi-denominational group Educate Together was significantly preferred in all five areas. There was strong support for all-Irish schools patron An Foras Pátrúnachta in Arklow (26%) and Tramore (21%), and one-in-five parents who wanted more choice in Tramore opted for the local vocational education committee.
Mr Quinn said while many parents were happy with the schools available in their area, there was a clear demand from others for more choice.
"I will now ask the main patron in each area, the Catholic bishop or archbishop, to consider the reconfiguration options open to him which would allow suffic-ient school accommodation to be made available to facilitate this choice," he said.
Mr Quinn wants their interim responses, based on local school consultations, within three months and a final response in six months.
Fr Michael Drumm, chair of the Catholic Schools Partnership, which represents bishops and religious orders, said the responses show 5% to 10% of parents want alternative schools for their children. His interpretation is based on an assumption that those who did not respond do not want any change to current school choices.
"There’s clearly a huge interest in Catholic schooling but there is also a clear need to reconfigure the system for the minority of parents who want additional forms of patronage. Those who told us the level of demand was 50% were clearly wrong," he said.