Begovic has been strongly linked with a move to a Premier League heavyweight during the winter window, with the likes of Manchester United and Arsenal said to be keen on acquiring his services.
Stoke were said to be lining up a move for Birmingham City's Jack Butland as cover, but there has been no movement on the goalkeeping front at the Britannia Stadium.
Begovic does not see that situation changing any time soon, with there no intention on his part to take on a new challenge elsewhere.
He told Sky Sports News HD: "I am in for training today, so I'm looking forward to staying and looking forward to the game at the weekend.
"I don't see anything happening now, that time has passed. I'm just concentrating on what I need to do here and then I'll have my popcorn ready to watch you guys tonight to see what happens."
On the Butland rumours, Begovic added: "I would look forward to it if he comes. He's another great talent.
"We have a good group of goalkeepers already here and it would be good to have the competition around. He's a big talent and I'm sure he would make us better."