In a joint letter to the Observer, they urge ministers to reconsider the plans.
The LGA says the plan to remove all schools from council control has caused "enormous concern" .
But, on Saturday, Education Secretary Nicky Morgan insisted there was "no reverse gear" on the changes.
Speaking at the NASUWT teacher union conference in Birmingham, Ms Morgan said she would not row back on the government's reform agenda, saying the plans would put control over schools into the hands of heads and teachers.
But this did not deter the National Union of Teachers, gathering for its annual conference in Brighton, voting for a ballot for a one-day strike next term over the government's plans.
The Observer letter, signed by Conservative councillors as well as those from opposition parties, says: "There is no evidence that academies perform better than council maintained schools.
"Where a school is failing, there is no question that action must be taken - but converting every school, regardless of performance, to an academy will not tackle those issues."
It goes on: "Schools value the option to become academies - and the support they receive from their local councils to do so - where they believe this is in the best interests of their students and communities.
"Forcing the change upon every school goes against, in many cases, what parents and teachers want, and there will be a large financial implication for local authorities at a time when communities are already suffering the impact of significant budget cuts."
The councillors continue: "We urge the government to listen to the concerns of families, teachers, unions, politicians and experts and rethink the proposals in the White Paper."
The letter is signed by Cllr David Hodge, Leader of the Conservative Group, Cllr Nick Forbes, Leader of the Labour Group, Cllr Marianne Overton, Leader of the Independent Group and Cllr Gerald Vernon-Jackson, Leader of the Liberal Democrat Group.
Leaders of the Conservative, Labour and Liberal Democrat groups in the Local Government Association have combined to oppose plans to force all English schools to become academies.