Al-Qalam primary school in Glasgow wants public money to help it expand into secondary education, but it wants to remain independent from council control.
The plans would see the 60 pupil school funded by tax payers but controlled by governors.
State funded Catholic schools, and schools of other faiths in Scotland, are currently run by local authorities.
Michael McGrath, director of the Scottish Catholic Education Service told the Herald the Church would have no objection to other faith schools but he suggested they had to be under council control.
He said: "Our Church obviously believes in the value of denominational schools and would be supportive of other wider forms of denominational provision being available where there is public demand.
"Of course we would expect these schools to be managed and governed in the same way as Catholic schools are as well as being regulated in the same ways."
The proposal has provoked debate about faith schools.
Nabeel Shaikh, general secretary of the Glasgow Central Mosque, told the Herald there was increasing demand for a Muslim state school: "Lots of questions are raised over why there isn't any state-funded Muslim school. I live in East Renfrewshire which has great Catholic schools and where a new £13 million joint campus for Jewish and Catholic pupils.
"Considering the number of Muslim pupils around Glasgow, many ask why we don't have a faith-based school but where the national curriculum is taught. Pupils could freely practice their beliefs but the curriculum is predominantly secular."
The school proposes to be "non-selective" and open to all faiths. It said it wanted to specialise in Islamic education.
The plans will now be considered by Scottish First Minister Nicola Sturgeon.