The Local Government Association is to table an amendment to the Bill, which would place a statutory duty on schools to promote the attainment of young people in care. Currently only local authorities have a duty to do so.
A spokesman for the Department for Education and Skills said: "Schools have a critical role to play, but the key responsibility for the educational achievement of children in care lies with the local authority which looks after them. It is the local authority which takes on the role of parent on behalf of the state."
The move by the LGA is being backed by the National Children's Bureau charity.
Under the proposals, each school would appoint a governor to look after the interests of children in care on its roll. A 1999 TES survey of the educational neglect of children in care showed that two-thirds of councils did not know what these youngsters had achieved in national tests, and two-fifths had no idea about their attainment at GCSE.
Pupils in care often change schools several times as they move between placements. They are more likely to be excluded from school and fewer than one in 10 achieves five or more A*-C grades. They are often placed in failing schools, where difficulties are exacerbated, the TES study found.
A quarter of the prison population and up to a third of rough sleepers have been in care. The underachievement of this group has been known about for more than 20 years. In the early 1980s it emerged that 11-year-olds in care were two years behind their peers in English.
The Bill says that local authorities' duty to promote the welfare of a child includes "in particular a duty to promote the child's educational achievement".
Paul Ennals, the NCB's chief executive, said: "Anecdotal evidence suggests that some schools are resisting taking looked-after children at short notice.
"There is also great variability in the quality and availability of support for this group in schools."
Mr Ennals said NCB research showed that while children in care identified teachers as the most common source of academic and pastoral support, some believed teachers hindered their progress.
Alison King, chair of the LGA's social affairs and health executive, said:
"We are seeking an amendment to make sure that schools are also required to take responsibility for ensuring that looked-after children are supported to reach their full potential at school.
"We propose that each school governing body appoints a governor with these responsibilities."