Reports have suggested that Manchester United and Liverpool are interested in the Bosnia international, who, as an ever-present for the Potters in their 15 Barclays Premier League games this term, has conceded only 12 goals and kept seven clean sheets.
Begovic was quoted as saying he is happy to stay and Pulis stressed his determination to keep hold of all his star men for the rest of the season. He said: "If a big club comes in for any of our players, then it is very difficult to say 'no, we are not going to do it.'"
Pulis added: "You can talk as much as you want, though - one thing I will say is that it is a big year for us this year and we will not be selling any of our better players in the January window, and that is full stop. It is a massive year for every club in the Premier League and this football club realises and recognises that.
"You hope you get interest because it shows the player is playing really well. But there will be no sales. We will be looking to bring one or two in, but there will be no sales of any of the big players at this football club during the window."
Begovic had been quoted as saying: "As far as I know it's all speculation. It's always nice when the biggest teams in the world are looking at you.
"It's a compliment, but it doesn't mean anything.
"We keep our feet on the ground here and the manager wouldn't let that change.
"I have three-and-a-half years on my contract and I am concentrating on playing and staying in the team and trying to help the team be successful. I am happy here so there are no issues with that."
The 25-year-old joined Stoke from Portsmouth in 2010.