Reports this week had suggested City were keen on signing Begovic, with Joe Hart potentially moving in the other direction on loan.
Begovic has been Stoke's star performer over the last 18 months and was heavily linked with a number of the Premier League's big guns in the summer.
Coates told the Stoke Sentinel: "This latest story was certainly news to me and I suspect it is just a bit of speculation. That's my guess, anyway, and we have certainly had no approach.
"He is a top goalkeeper, so this kind of thing is going to happen. We are just concerned that he concentrates on playing for us and doing what he has been doing up to now.
"It is important we keep him for the rest of the season. It is an important season and he is a key player. He certainly hasn't made any noises."