Earlier this year, the council voted to rent its land to mobile phone companies for masts, provided they were on non-contentious sites.
Contentious sites were identified as those close to schools, nurseries, open space near playgrounds or residential property.
Rosebank and Waterside councillor Jack Young is concerned that a mobile phone company has contacted the council about siting a mast next to Merkland playing fields.
He said: "It would be hypocritical of me to criticise mobile phone masts, because I use a mobile phone.
"I know the council made this decision because suitable land is becoming very scarce. However, I am extremely unhappy that an approach has been made by a private company to site a mast near the running track at Merkland Outdoor Recreation Centre.
"This facility is widely used by many young people and has only just been upgraded to contain all-weather football pitches.
"Since finding out about this I have made arrangements to meet with senior sports facility officers at the council to express my concerns.
"A petition is also being organised by constituents and will be submitted to the council's chief executive."
A spokesperson for East Dunbartonshire Council said: "There is an established, council-approved procedure for dealing with enquiries from telecommunications operators who approach us with proposals to put their equipment on council land.
"This procedure is currently being followed in relation to the approach by 3G for the Merkland site. No decisions have yet been made.
"Should the proposal be submitted as a planning application, the council's planning policy on telecommunications development will be adhered to."