The highly-rated Butland was signed from Birmingham in January 2013 when previous manager Tony Pulis was still in charge, but the 20-year-old was immediately loaned back to Birmingham for the rest of the season.
Begovic has been Stoke's number-one stopper for much of the last three years, but was thought to have interested Liverpool prior to their signing of Sunderland keeper Simon Mignolet.
But having started last season by winning his first senior England cap, in a friendly against Italy, Butland wants to begin this campaign by claiming a regular place in Hughes' starting line-up.
"That's the aim," the 20-year-old told Stoke's official club website. "I know that won't be easy because Asmir is a top, top goalkeeper and has done really well for Stoke City in the past but I'm excited about the challenge.
"I'm settling in very well. I came in a few times in the second half of the season which helped me get to know the lads and made it a bit easier when we reported back. I'm getting on with the lads and enjoying the training."
The former Cheltenham player is also keen to put England Under-21s' disappointing European U21 Championship firmly behind him, having played in two of the three defeats that sent Stuart Pearce's side home from Israel after the group stage.
Butland added: "You always have an opportunity in football to put things right pretty quickly and that's what I can do this season, which I'm really looking forward to doing."