The Bosnian international moved to the Britannia Stadium in January 2010 from Portsmouth in a deal worth £3.25 million, and has since displaced Thomas Sorensen as the club's No.1.
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Despite the speculation, however, the 25-year-old insists he is focused solely on performing at the highest level for his present employers.
"I have no time to think about this. My only concern is Stoke City as I want to be as good as I can be for my club, and for my national team," he told Sport Sport.
"So quite simply, I really don't have any time to think about a move or anything else."
But while he remains unwilling to engage with speculation over his future, Begovic admits he is flattered to be linked with the likes of United and Liverpool.
"I must confess it is nice to hear these things," he added. "It is always nice to hear what others are saying, it means that you are doing a good job."