After the Potters snapped up young England goalkeeper Butland, Begovic was thought to be on his way out of the club, with Liverpool, Arsenal and Manchester United interested.
However, as the transfer window has developed, any talk of the Bosnian stopper moving on has died down, which has allowed him to concentrate on getting ready for the new season.
"I always tried to stay focused and not concentrate on outside things," Begovic told The Sentinel.
"You never know for sure in football what can happen, of course, but as far as I am concerned, I am happy things are settled.
"I am happy to stay and I am not looking for a move. It's where I see my future."
Stoke chairman Peter Coates confirmed that there have been no bids for the 26-year-old, and indicated Butland may be loaned out to try and get him more regular first-team football.
"Begovic will be our goalkeeper at the start of the season," Coates told The Sun.
"And I would think we'd be definitely looking to get Jack Butland playing somewhere next season where we can improve his game."