Forecasts by the agency - the quango which administers finances for the grant-maintained sector - show growth rates in different areas may be as low as 5.8 per cent or as high as 24 per cent.
Its claims are based on an inquiry into the planning of secondary-school places in Bournemouth, Derby, Luton, Milton Keynes, Poole, Portsmouth, Rutland, Southampton andSwindon.
These are the nine unitary authorities, established this April, with which the FAS shares responsibility for planning school places.
Its report discloses that by 2003 the number of 11 to 16-year-olds will increase by almost a quarter in Portsmouth, which has one of the highest population densities in Europe.
In Swindon, where there are four grant-maintained schools and seven local authority-controlled secondaries, the number of pupils will rise by 21 per cent.
The lowest increase will come in Luton, where figures are predicted to rise by less than 6 per cent.
All authorities have surplus places, but it is expected that overall supply and demand will be more or less in balance by 2003.
Nevertheless, some authorities will have particular problems. Derby is expected to have a surplus of 1,370 places (8.5 per cent) while Portsmouth will need an extra 1,790 places (20 per cent).
In Derby the FAS predicts that, despite the surplus places, in some parts of the city parents will find it increasingly difficult to obtain a place for their children in the school of their choice.
The FAS also acknowledges that it will need to have discussions with Portsmouth council on how to meet the rapid growth in secondary pupil numbers.
It says that in Bournemouth a programme of school expansion must be developed which takes account of the balance of supply of places in the grammar and secondary schools.
In Rutland the FAS and local authority need to develop both policy and provision for special needs pupils.