A motion to the council's executive from Marilyn MacLaren, a leading Liberal Democrat councillor, argued: "Teachers come into contact with a lot of people - and their viruses. They are on the front line. When they go off sick, a significant number of supply teachers are needed to take their classes at considerable cost to the education authority.
"Pupils' education, too, is at risk of being disrupted."
But Colin MacKay, secretary of the Edinburgh local association of the Educational Institute of Scotland, said: "This smacks of the nanny state. I cannot see teachers lining up in the staffroom to get inoculated against flu.
"Teachers can access flu jabs through their own GPs. But if flu jabs are to be offered on school premises, what about inoculations against hepatitis and tetanus? Where would it end?"
Mr MacKay says 3.5 per cent of teacher working days are lost through sickness each year. "This is the lowest rate of all council employees," he said. "If this is a service an employer should offer, then all employees should benefit."
Thousands of teachers, support and clerical staff in more than 170 nursery, primary, and secondary schools could be offered the jab which would be administered by Lothian Health staff.
But nothing will be done until officials carry out a study on the feasibility of the move.