The Church of England is struggling to find enough Christian headteachers to lead its 4,700 primary and secondary schools.
The 4,500 primary schools and 200 secondary schools, which are among the best in the country and are highly sought-after by parents, are having to recruit from other faiths or none at all to fill the posts.
The Church of England has a policy that headteachers in its schools need not be church members but must be "on board" with Anglican values. This is in contrast to the Catholic Church where headteachers in its schools must be practising Catholics.
In the "needs analysis" report, disclosed in the Daily Mail, the Church's education office warns that the increasingly elderly body of headteachers means effort has to be made to recruit new, strong leaders.
About a million children attend CofE schools. The Church is the biggest sponsor of academies in England. More than 500 independent schools are also CofE.
The document says: "Recruitment of school leaders with the necessary understanding and commitment is proving increasingly difficult, and sometimes impossible. Many dioceses have become more flexible around the requirement that headteachers need to be practising Christians and can reference successful church school heads who are from other faiths or none at all but are able to maintain a clear vision for education."
It warns of a risk to the Church's vision of education if enough teachers and school leaders with a deep understanding of and engagement with the Church of England cannot be deployed.
There was also a perception from outside "that it may be more difficult" to be a headteacher in a CofE school. The problems were particularly severe in rural dioceses such as Exeter and Norwich.
Rabbi Dr Jonathan Romain, of the Accord Coalition, which campaigns to end religious discrimination in school staffing and admissions, said: "The growing number of Church of England schools that are appointing senior staff from outside the faith is encouraging."
Rev Nigel Genders, CofE chief education officer, said: "Church schools continue to be oversubscribed and popular with parents and pupils, opting for a Christian based education whatever their own faith. Both community and church schools increasingly testify to difficulties in recruiting headteachers and our recent consultation has shown a strong desire for more support in training new leaders. Heads and teachers have told us that they want more help and better training to enable them to promote the Church of England's vision for education. To this end we are consulting about plans to better equip and support leaders and teachers across the country in a fast-moving educational environment."