The highly-rated 26-year-old was linked with various top European sides over the close-season, but remained at the Britannia Stadium.
Although Stoke lie 16th in the Premier League following a run of three straight defeats, Begovic is satisfied he made the right decision in staying put.
However, he admits to being flattered by the interest from afar, which could well intensify once again in the January transfer window.
"I'm happy because I stayed at Stoke City," Begovic 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.
"The interest motivates me even more and I would really like to become an even better goalkeeper.
"I'm a very ambitious guy but, then again, I'm pleased with the club where I'm currently playing."
Begovic is on international duty with Bosnia-Herzegovina, who face Liechtenstein and Lithuania in their final FIFA World Cup qualifiers.
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