Glasgow has warned ministers against pulling teachers out of the classroom to train as headteachers and wants to extend the teaching year by five days to create extra time for staff development.
"There are serious implications for teachers' conditions of service, although many teachers already undertake study in holiday periods," the city states.
It also advises the Scottish Office that aspiring teachers with the proposed "suite of qualifications" - certificate, diploma and master - should not automatically assume they will gain promotion. Moray, meanwhile, says "proper" qualifications and training for headteachers are "long overdue".
Moray says heads in post should not have to take the qualifications and should only receive training if they move to another school. But they should be encouraged to take up staff development opportunities.
Glasgow suggests the qualifications should become mandatory over time and that the Scottish Office should review its advice on appointment procedures since "school boards do not always give due attention to the personal professional development of applicants".
The council supports mandatory qualifications for heads and recognises it will take several years to move from a Certificate to a Master. More controversially, the council backs compulsory appraisal of all teachers to help in identifying existing competences. It adds: "There is a clear case for separate funding to be identified and ring-fenced to ensure a reasonable level of equity across the country."