The Bosnia-Herzegovina international, who represented his country at the FIFA World Cup last month, has been the subject of speculation linking him with a move away from the Premier League club - with Real Madrid rumoured to be among his suitors.
But Coates, who saw manager Mark Hughes guide Stoke to a best ever Premier League finish of ninth last season, is eager to keep hold of the 27-year-old.
"He's very much our player and nothing has happened recently to change that because we haven't had any inquiries for him," he said in quotes reported by 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, who moved to the Britannia Stadium from Portsmouth in 2010, has two years remaining on his contract.
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