The 25-year-old impressed last season, despite a difficult campaign for Stoke, leading to reported interest from Manchester United and Liverpool.
However, Begovic says he is happy at the club and is looking forward to the next campaign, while he is also eager to work under new manager Mark Hughes, who has replaced Tony Pulis.
|10/3||Stoke are 10/3 with Coral to be relegated
“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 so I can't really complain."
Stoke are well stocked in the goalkeeper department having signed Jack Butland from Birmingham in January and with Thomas Sorensen also in the squad, and Begovic has been full of praise for his fellow shot-stoppers.
"Jack [Butland] is a good up and coming keeper and everyone knows about Thomas [Sorensen] and the qualities he has," he added.
"We have a strong group of goalkeepers and I am looking forward to being part of it."