The highly-rated Bosnian put in another standout showing over the weekend as the Potters held Hull City to a goalless draw at the KC Stadium.
That performance has helped to fan the transfer flames, with the likes of Chelsea, Manchester City and Arsenal having reportedly been closely monitoring his situation for some time.
Stoke have successfully fended off the interest thus far, but speculation is likely to intensify again heading into the winter window.
Begovic accepts that he can do little to stop gossip from circulating and maintains that he is not pushing for a move away from the Britannia Stadium.
The 26-year-old shot-stopper said: "I don't pay much attention to it.
"I take it as a huge compliment if people are saying nice things and paying attention but there's a pressure to keep performing so I have to concentrate on my job each and every day.
"I've got two-and-a-half years left on my contract, so I'm pretty tied down and pretty settled, so there's no situation there.
"I feel comfortable with where I am and the management. The coaches are great and that helps me perform better.
"Of course everyone wants to play at the highest level as possible. I'm no different.
"I think we can all achieve great things here in the future."