Begovic joined Stoke from Portsmouth during the winter transfer window 2010 and was targeted by Chelsea - and Madrid boss Carlo Ancelotti - in the summer of the same year.
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"There is no truth as far as I'm aware," he told Sky Sports News. "I've had no contact with either club, so from my point of view it's flattering to hear those kind of words.
"Like I said, [they are] two [of the] biggest clubs in the world and who wouldn't be flattered if you mentioned them in the same breath as those two kind of clubs?
"I'm just enjoying playing my game here, I focus on my job each and every day [and] make sure I perform each and every weekend for Stoke City.
"I've enjoyed my time here and I don't want that to change, but like I said everyone has ambitions and try to go as far as they can in the game, and it's flattering to hear the big clubs [linked to you].
"But as far as I'm concerned nothing is concrete, there has been no contact and I just enjoy doing my work each and every day here."
When asked specifically about Madrid, Begovic added: "My ambitions are to play at the highest level of the game and test myself against the best players in the world.
"Qualifying for the World Cup this year with my national team [Bosnia] has been a huge achievement and that's something I'm really looking forward to and it's just about progressing and developing in the game.
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"I try to work hard each and every day to progress in my game, to work hard at my game and get better as a goalkeeper and of course I want to play at the highest level possible."