Robin Squire, schools minister, wants a primary CFF to be piloted in Essex, Gloucestershire and the London borough of Hillingdon.
He also intends the CFF to be introduced in the new unitary authorities of Bournemouth, Derby City, Luton, Milton Keynes, Poole, Rutland and Thamesdown.
The CFF is likely to be extended too in the metropolitan borough of Trafford, once a bastion of Tory rule but now controlled by Labour.
The Department for Education and Employment is consulting on the proposals which would come into effect from next April.
Detailed arrangements for the CFF vary between authorities but are based on a core of factors - age-weighted pupil numbers, a fixed-cost sum, free school meals and special needs.
The CFF was first introduced in April 1994 and ministers say it represents the first stage of a system that will eventually break links between GM funding and LEA decisions on budgets and local management of schools schemes.
They now want to introduce a national funding formula for the GM sector - but opted-out heads insist that such a scheme should be extended to all schools.
Copies of CFF for Secondary Schools: proposals for coverage in 1997-98 and A Common Funding Formula for GM Primary Schools are available from the DFEE Publications Centre, P0 Box 6927, London E3 3NZ. Telephone: 0171 510 0150.