In a briefing to councillors, head of schools Mike Rosendale instructed them to refer queries from school staff to headteachers - even if the employees are also constituents.
It comes after teachers themselves were warned not to speak to elected members about the impact of cuts facing secondary schools.
Councillor Paul Godzik, Labour's education spokesman, said he would continue to take queries and listen to concerns.
He said: "I think that teachers and councillors should be able to have an open dialogue.
"If someone came to me with any issue, I would hear that issue in confidence.
"I certainly wouldn't be referring it on to the headteacher if that individual did not wish me to do that.
"One of the important things that we as councillors do is take issues in confidence, hear people's complaints and act on them.
"I don't think this advice is in the spirit of that, and most councillors will not look on this favourably.
"The majority will continue to hear constituents' issues in the strictest confidence and I do think without the permission of the constituents to refer it to the headteacher, councillors could be getting in to data protection and confidentiality issues and would have to consider that very carefully."
The review of the management structure in schools focuses on reducing the number of deputy heads and cutting principal teacher positions.
Education chiefs want to create a "faculty model" in city schools, with principal teachers taking charge of a whole department instead of individual subjects.
It has been suggested that this might even include one teacher being responsible for a department that covers the vastly different subjects of English and modern languages.
Councillor Alison Johnstone, education spokeswoman for the Greens and Lothians MSP, said: "These cuts to senior jobs are a bad move for Edinburgh's education services, which is why Green councillors voted against them in February.
"The SNP/Lib Dem administration have not thought these changes through properly, and they have not properly considered the pressure this will place on schools.
"I understand that staff are contractually supposed not to raise their concerns with their local councillors, but I regard this as a profoundly undemocratic restriction."School staff who live in my ward have already spoken to me about the problems this move will cause, and I will guarantee anonymity to anyone else who wishes to discuss the situation with me."
A council spokesman said staff always have "access" to councillors.
He added: "Headteachers are fully aware of the process and of the proposed changes in their schools and are the best source of information for staff who may have questions to ask.
"Of course, staff always have access to their local councillors to discuss concerns."