The constitution said any plans for new legislation must take the financial and administrative effects on councils into account.
It added: 'The union shall respect the national identities of its member states, inherent in their fundamental structures, political and constitutional, including for regional and local self government.'
Critics say the constitution will weaken central and local government by more than doubling the European Parliament's powers, from 34 to 70.
Head of the bureau's office in Brussels Richard Kitt said: 'The draft treaty represents a significant step forward for local authorities in the way they deal with the EU. It will mean that we do not see any more fridge mountains emerging due to a lack of consultation with local authorities. It will also prevent the EU from legislating where it would be better for local authorities to regulate their own services.'