Peter Coates has hailed the goalkeeper and is keen for the 27-year-old to see out his career at the Britannia Stadium, before admitting there have been no offers for the players
The Bosnia-Herzegovina international, who represented his country at the World Cup, has been the subject of speculation linking him with a move away from the Premier League club - with Real Madrid linked.
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"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.