The 25-year-old goalkeeper has been in impressive form for the Potters this campaign, keeping seven clean sheets in the Premier League and conceding just 12 goals.
Red Devils boss Sir Alex Ferguson has been credited with an interest in bringing the Bosnian international to Old Trafford, as neither David De Gea and Anders Lindegaard have been able to nail down the number one spot.
Brendan Rodgers is also believed to be in the market for a new goalkeeper, with speculation suggesting long-term custodian Pepe Reina could leave Anfield in January.
However, both Manchester United and Liverpool's hopes of securing Begovic's signature have been dealt a blow after Pulis reiterated his desire to keep all first-team players at the Britannia Stadium beyond the New Year.
"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,'" he explained.
"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."
Pulis added that it is always a compliment to see Stoke's players linked with the Premier League powerhouses, but that doesn't mean that they will buckle under the pressure from England's leading clubs.
"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."