The goalkeeper has been touted as a transfer target for Premier League leaders Manchester United, while Manchester City, Arsenal and Liverpool have also reportedly shown interest.
Despite growing speculation that the Bosnian is keen to complete a move to the Red Devils, the 25-year-old has now stated that he is likely to remain at the Britannia Stadium and is looking ahead to the rest of the season.
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"I don’t see anything happening now. I think that time’s passed. I’m just concentrating on what I need to do here."
Potters boss Tony Pulis has been linked with a move for Birmingham keeper Butland, with reports suggesting the club are poised to make a formal bid after the 19-year-old rejected Chelsea due to the prospect of no first-team football.
Current number one Begovic insisted that the Englishman's arrival would be great for the club and would only add to the healthy competition for the starting position in between the goalposts.
"[I'm] looking forward to it," he added. "If he comes in he’s another great talent.
"Obviously we have a good group of goalkeepers already here and it’ll be good to have the competition around. I would welcome him if he comes, for sure, he’s a big talent and I’m sure he’d make us better."
Stoke, however, face stiff competition for the youngster's signature with Southampton and Everton both failing to land the player in the summer, and with Liverpool, Fulham, Newcastle and Premier League reigning champions City also on alert.