The 27-year-old, who is widely regarded as one of the best stoppers in the Premier League, continues to be linked with a move away from Stoke this summer.
However, Coates is confident that the former Portsmouth goalkeeper will remain with the club.
"He's very much our player and nothing has happened recently to change that because we haven't had any inquiries for him," Coates told the Stoke Sentinel.
"My own preference is that he stays. He can stay forever as far as I'm concerned. He's clearly a top-class goalkeeper, but also a very good sportsman and a lovely man.
"I also think he himself is very settled here and genuinely enjoys being at this football club."
Begovic has two years remaining on his current deal.