A Scots teaching union has questioned the role of councils in running schools and managing education provision.
The EIS said a reduction in the number of education authorities should be considered to cut costs.
The union said schools could instead be governed by dedicated education boards, which it said would be best placed to deliver Scottish government priorities.
The education minister said he welcomed ideas on how to deliver services more efficiently across council boundaries.
However, Mike Russell also said the Scottish government had no plans to reduce the number of local authorities, which currently stands at 32.
His comments came after the EIS's general secretary, Ronnie Smith, said education structures should be examined at a time when councils were looking to share services to cut costs.
Mr Smith said: "We live in difficult economic times, and all public services face the challenge of how to continue to provide a high level of service in a more cost-effective manner.
"In this context, it is right that we ask questions about the need for 32 individual local authorities to deliver national education policy, operating schools and providing education for our young people."
He said sharing services in administration, procurement and human resources could "save a considerable amount of money and allow the vital frontline provision of learning and teaching to be protected".
Mr Smith suggested a structural revamp would not only save money, but would also ensure the Scottish government's priorities were more likely to be met.
He said: "The EIS believes that all too often important national policies have become diluted at a local level as local authorities identify their own policy priorities which are not always consistent with national policy."
"With past experience of the difficulty in delivering national priorities through the concordat with local authorities, it is worrying that the Scottish government has again struck a budget deal with Cosla [council umbrella body] which provides a shopping list of desired government priorities without any levers to ensure that they are delivered.
"As the concordat proved, you cannot seek to deliver such national priorities while at the same time promising more freedom to councils to set their own priorities on education."
He added: "Education boards, focusing purely on education, would have a clear responsibility for school education at a local level and could have clear benefits in terms of ensuring consistent delivery of nationally agreed educational priorities and policies.
"However, it would be equally important that there was proper democratic accountability within the boards so that local views and priorities are not overtaken."
The suggestion prompted an angry response from Cosla, the body which represents Scotland's local authorities.
Councillor Isabel Hutton said: "There is a debate to be had about how the future shape of the public sector meets the financial challenge, but this sort of one dimensional, simplistic, off-the-shelf analysis will not get us very far.
"There are very good reasons why education should continue to be a local authority service, not least because of this year's record exam results and the support of international bodies like the OECD."
Mr Russell said: "As I've made clear, we face tough choices about what we spend our resources on.
"We need to prioritise and I want to be absolutely clear that frontline services should be protected by targeting reductions at bureaucracy that does not contribute directly to service delivery.
"I shall also welcome and support ideas that will deliver education services more efficiently through more effective deployment of resources across boundaries - such as those in Stirling and Clackmannanshire, and East Lothian and Midlothian."