Press Association Sport understands the Potters are interested in Butland, the 19-year-old England international who has been linked with various Barclays Premier League clubs.
With reports on Wednesday suggesting Butland had turned down a move to Chelsea because he did not want to sign for a club where he would not have a realistic prospect of playing regular first-team football, news that Stoke are looking at him would appear to raise questions about the future of their current first-choice stopper Begovic.
The Bosnia international has been linked with the likes of Manchester United and Liverpool, but he told Sky Sports News on Thursday: "As far as I know (I am staying at Stoke), yes.
"I'm in for training today [Thursday] so I'm looking forward to staying and looking forward to the game at the weekend (Saturday's league trip to Arsenal).
"I don't see anything happening now. I think that time's passed. I'm just concentrating on what I need to do here."
On the prospect of Butland moving to Stoke before the January transfer window closes at 11pm on Thursday evening, Begovic added: "(I'm) looking forward to it.
"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, who are not commenting on reports they have lodged a formal bid for Butland, also have goalkeepers Thomas Sorensen and Carlo Nash on their books.
Meanwhile, the Potters remain hopeful their proposed signing of United States winger Brek Shea from Major League Soccer outfit FC Dallas will be completed before the deadline.
The Staffordshire club announced earlier this week that they had reached agreement with Dallas over the transfer of the 22-year-old, who has passed a medical and took part in Stoke's training session on Thursday morning.
There are still work permit issues to be concluded, though, with an appeal hearing on the matter taking place on Thursday afternoon.
Another player who trained with the Potters on Thursday is striker Peter Crouch, who they had received an enquiry from QPR about.
While it is unclear whether or not there will be movement on that front, one definite departure from Stoke on Thursday is midfielder Rory Delap, with the club having confirmed that the 36-year-old has joined Barnsley on loan until the end of the season.