They have complained that the move in the London Borough of Tower Hamlets, where the British National Party won a council seat in 1993 but lost it again in 1994, could be racially divisive.
Many Muslim parents withdraw their children for Eid and some schools close for it, but there are fears that the new compulsory holiday will upset non-Muslim parents.
Of 25 head teachers who replied to the council's consultation on the issue, all but one said it should be left to their discretion. They also pointed out that because Eid happens the day after the new moon can be seen over Mecca, its date is not known for certain until immediately beforehand.
However, councillors have insisted that schools must close for two separate days for the two festivals of Eid, which are two months apart and which will fall in the late spring next year, and that the summer holidays will be two days shorter. Christian voluntary aided schools will be requested to close, but it seems likely that many will refuse.
Some other London boroughs close their schools for non-Christian religious holidays: Waltham Forest has days off for Eid, for the Hindu festival of Diwali and for Guru Nanek's birthday, which is celebrated by Sikhs.
Half the primary school children in Tower Hamlets are Bangladeshi Muslims, but one-third are English, Scottish or Welsh and almost one in five are from a wide range of other racial backgrounds.
The National Association of Head Teachers (NAHT), which represents 90 out of the 100 Tower Hamlets heads, says the move will be counterproductive. It has written to the Department for Education and Employment about it, but has been told that holidays are the responsibility of the local authority.
Heads whose schools close for other non-Christian holidays at the moment are upset because they will no longer be able to do so.
Michael Russell, head teacher of Malmesbury Junior School in Bow and NAHT national executive member for the borough, said that in an area of rich cultural diversity, it seemed narrow to grant a Muslim holiday but to ignore other racial groups.
"It would be a great shame if this was divisive and if it increased racial tension in an area where the local authority has already done so much to promote harmony, trust and understanding," he said.
A council spokeswoman said the holiday was designed to allow all pupils to take part in a festival already enjoyed by many.
"It helps to promote a multicultural environment for all our pupils to learn from. The main holidays are tinkered slightly in order to ensure that no valuable school time is lost," she said.
A spokeswoman for the Department of Education and Employment said: "It is for local education authorities and governing bodies to decide the arrangements that best suit them, in line with local circumstances."Reuse content