One of the main themes of the controversial schools white paper, published in October, is to create more choice for parents and pupils.
But Alison King, LGA education spokeswoman, said it would have the opposite effect, warning of free-for-all admissions that would see parents and pupils "lost in the maze".
The Conservative councillor from Norfolk acknowledged the pound;12 million the Government had pledged to pay for choice advisers to guide disadvantaged parents.
But she said any benefits would be lost because of the greater independence schools would have over admissions. "Fairness cannot exist if schools are allowed to pick their pupils," said Mrs King.
The LGA appeared before MPs on the Commons education select committee this week and argued for the admissions code of practice which bans selection and admissions interviews, to be made statutory. Ministers are understood to be ready to concede this to stave off a Labour backbench rebellion. Ruth Kelly, the Education Secretary, chose not to rule out such a measure when answering questions in Parliament.
The TES understands ministers are also prepared to make a U-turn over plans to ban the creation of new community schools, in an effort to convince waverers that next year's Education Bill will not signal the beginning of the end for local authorities' role in education. Government insiders still hope to push through the Bill without concessions but admit this is unlikely.
There was more upheaval this week as Richard Darlington, Ms Kelly's chief spin doctor who oversaw the launch of the white paper, left to return to his old job at the Institute for Public Policy Research.
The strength of feeling over the White Paper was shown at a recent meeting of junior ministers when even normally loyal figures such as Yvette Cooper, minister of state in the office of the Deputy Prime Minister, spoke out against the plans. The result of the Conservative leadership election, expected on Tuesday, may also have an impact.
David Cameron, shadow education secretary and favourite, largely supports the proposals but rival David Davis has promised to do all he can to hasten the end of Tony Blair's premiership.
Further criticism of the plans has come from Ron Glatter, a visiting professor at the Open university and former director of the Centre for Education Policy and Management.
In written evidence to the select committee he said they appeared to be based on assertion rather than evidence and were a "risky strategy in terms of the objectives of greater social equity, overall attainment and parental satisfaction".