The Bosnia and Herzegovina international still has two years remaining on his current deal but he has regularly been linked with a move away from the Britannia Stadium after reportedly impressing a number of other clubs with his performances in recent seasons.
Chairman Peter Coates has previously spoken about the possibility of a new deal for the goalkeeper, saying he could ‘stay forever as far as I’m concerned’.
Begovic has yet to discuss his future with the club but is staying relaxed about the situation.
He told The Sentinel: "We haven't actually spoken yet, but that will probably happen in the next little while.
"With the World Cup in the summer, there really wasn't time to sit down and focus on that kind of thing then.
"Now everyone's back and the season's about to get going, so we'll see what the future holds.”
Stoke are looking to push on under Mark Hughes this season and have made some exciting signings, including Bojan Krkic from Barcelona.
Begovic is ready to play a key part after insisting that it did not cross his mind that he would be leaving this summer.
"The plan was always to come back, although I won't make promises I can't keep of course," he added.
"But I'm excited about the plans that the club and the manager have for the future.
"I want to be part of it, and the new signings show the club's vision and how they want to move forwards. There's plenty of room to grow.
"Every player wants to be part of a successful team and the players brought in here, not just Bojan, have been exciting.
"We want to push on now and have another successful season in the Premier League."