Liverpool are reportedly targeting Begovic, who is out of contract at the end of the 2015-16 Premier League season.
Brendan Rodgers would have been alerted to Begovic's exploits at the Britannia Stadium on Saturday as Stoke stunned Arsenal 3-2.
But the Bosnia-Herzegovina international keeper said he is settled in Staffordshire - a place he has called home since joining from Portsmouth in 2010.
"My future is here at Stoke as far as I'm concerned. We've got a lot of games coming up and I'm hoping to be a big part of that," Begovic told Sky Sports.
"I’ve been here a few years now and enjoyed my time. I've really settled in and had a lot of support from the staff.
"We all have ambitions and goals. I work really hard to get better and try to improve as a player, and hopefully test myself at the highest level possible level in the future. Who knows whether that opportunity will present itself?
"We'll see what the future holds but I'm not thinking too far ahead."
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