Councillors will decide later this month whether Broomhouse and St Joseph's primaries, which currently share a site, should swap buildings or form a shared wing to cater for the influx of pupils choosing to go to the Catholic school.
Despite having a smaller roll and a bigger building, Broomhouse parents are strongly against giving up their larger school to St Joseph's and instead believe offering some extra classrooms to the Catholic school to create a shared wing would be the best option.
But St Joseph's parents find this unacceptable and say it would lead to "significant deterioration" in the provision of Catholic education to its pupils.
Along with the school, the Archdiocese of St Andrew's and Edinburgh has written to members of the education committee – who will take the final decision on 27 April – to warn them that if they choose the shared wing option, they could take the matter to Scottish ministers for breaching legislation.
They point out that the 1918 Education Act gave the Scottish hierarchy on behalf of the Catholic community "certain safeguards". The letter states: "These included a statutory right of appeal to Scottish ministers if the Scottish hierarchy considered that any proposals on school provision would lead to any 'significant deterioration' in the provision of Catholic education.
"We would contend that the proposed option two (to create a shared wing] could indeed lead to 'significant deterioration' in the provision of Catholic education for the pupils of St Joseph's Primary School."
The letter also highlights concerns over the practicalities of managing a shared wing, as well as the difficulties associated with retaining identities.
St Joseph's, for example, would want the shared wing to be blessed, and there are fears about how Broomhouse parents would react to this.
The letter continues: "The headteacher of St Joseph's would always strive to ensure that the identity, ethos and values of her school were upheld.
"Uniform, decor and iconography are all part of this.
"St Joseph's Primary School would wish to display holy pictures and statues throughout its accommodation."
Broomhouse parents have been campaigning to retain their school and have collected hundreds of signatures, as well as staging a protest. They are appealing to councillors to vote for the shared wing option.
Councillor Ricky Henderson, Labour's education spokesman, said: "In light of the fact that the St Joseph's Primary School community and the Catholic church has raised the reference to legislation, I have requested that the council solicitor or somebody with expertise in the relevant legislation be present at the special meeting because as councillors we have to be confident that decisions that we take are in compliance with the legislation."
A spokesman for the council said: "This is a sensitive issue and we are obviously very aware of our legal responsibilities as an education authority.
"Our aim is to provide a solution that works in practice.."