The government rejected the LGA amendment, tabled by crossbench peer Earl Listowel, at the bill's committee stage, but the association is looking to raise it again at the forthcoming report stage.
The bill contains a duty on councils to promote the educational achievement of looked-after children. But councils, and charities like the National Children's Bureau, have called for a similar obligation to be placed on schools.
An LGA spokesperson said: 'It is about making looked-after children a priority for the school so they are not treated like other children.'
In a separate development, children's minister Margaret Hodge told MPs that inspection and the threat of government intervention will ensure councils join up their children's services.
She told the education and skills select committee that departments would have an incentive to work together, as otherwise they would fail inspections and score poorly in the comprehensive performance assessment.
And she warned the Children Bill provided tough powers to intervene where children's services were judged to be failing. She said the government had built on the experience of intervening over education, which she claimed had helped some councils.
But she resisted pressure from members to agree to impose a new structure on councils.
'I know it would have been deemed an enormous intrusion in local authorities' ability to manage themselves. I would feel, as an old localist, wrong if I told them they have to do it in a particular way,' she stressed.
'This is one of the best bits of news local government has had for a long time - p utting themselves in the driving seat to create these new structures and environments.'