Begovic has made 100 first-team appearances since arriving at Stoke in January 2010, leading to recent speculation linking him with Liverpool and both Manchester clubs.
But the 25-year-old is adamant he has no plans to leave this summer.
"I haven't actually spoken with the new manager myself, but there will be a time when we can look at my situation," said Begovic.
"But as far as I am concerned, nothing has changed with the change of manager.
"I am committed to the club until someone tells me otherwise. And I am sure that whatever happens, the club will do what is best."
Begovic, who is currently away with Bosnia ahead of Friday's World Cup qualifier in Latvia, is excited by the prospect of Hughes arriving at the Britannia Stadium.
"I think it's great for the football club," Begovic added.
"Mark Hughes has got a pretty good track record in management in the past and hopefully he can come into the club and take us forward again."