The Potters goalkeeper was one of the club’s most consistent performers throughout a difficult season which saw them narrowly avoid relegation to the Championship.
With Tony Pulis leaving the club last month, Begovic was reported to be considering a move away from the Britannia Stadium amid speculation linking Arsenal and United to the Serbian.
But Begovic insisted he is already looking forward to reporting for a pre-season training with Mark Hughes at the Stoke helm.
“I am happy to be here and I am happy to stay here,” Begovic said.
“I am looking forward to reporting back in pre-season, meeting the new manager and staff and cracking on with the new season.
“The most important thing is I am at a club where I am settled and I feel comfortable. Everyone has been really good to me and the chairman has been fantastic.
“I am going to have to work hard to keep my place and I am looking forward to that.
“Jack [Butland] is a good up-and-coming keeper and everyone knows about Thomas and the qualities he has.
“He has a good deal of experience and quality. We have a strong group of goalkeepers and I am looking forward to being part of it.”
Get enhanced odds of 18/1 on Andy Murray to win Wimbledon or 10/1 on Novak Djokovic when you open a Paddy Power account.