The highly-rated Bosnia goalkeeper was subject to interest from various top European sides and, despite admitting to being flattered by the interest, he decided to remain at the Britannia Stadium.
"I'm happy because I stayed at Stoke City," he told Mundo.
"I feel great here and I love my football here, but I can't hide the fact that it was nice to be linked with so many big clubs."
Begovic is keen to continue his development at Stoke this season and he believes the speculation will help him improve, while hinting that he may be open to offers in the future.
"The interest motivates me even more and I would really like to become an even better goalkeeper," he added.
"I'm a very ambitious guy but then again, I'm pleased with the club where I'm currently playing."
Begovic's Bosnia and Herzegovina side currently top World Cup qualifying Group G and look set to be involved in next year's competition in Brazil.