Liverpool and Manchester United have been linked with a move for Begovic, who has impressed for the Potters this season, keeping seven clean sheets and conceding just 12 goals.
And Pulis, while adamant there will be no sales in the new year, hinted at "one or two" new arrivals.
Asked about the speculation linking Begovic with a move, Pulis 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.'
"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 previously said: "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."
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