The Potters manager believes the Bosnian should have been awarded a place in the Professional Footballers Association Team of the Year ahead of Manchester United's David de Gea
Manchester United keeper David De Gea was voted into the team by his peers - with all 11 players coming from sides in the top-seven placings - but Pulis believes the Bosnian's performances warrant recognition.
Begovic has been linked with a move away from the Britannia Stadium during the season - with Manchester United, Arsenal and Barcelona all linked - but his manager is keen to retain his services, but understands the speculation.
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"Asmir has had a full run at the team this season and grown in stature because of it. There is no substitute for playing games and that's why I would always encourage youngsters at this football club to go out and play league football to gain experience."
Stoke reach the 40-point mark on Saturday and are virtually guaranteed Premier League safety following back-to-back victories over QPR and Norwich, but Pulis insists his players will not be taking it easy in their final three games of the season.
"We've got 40 points, which is the benchmark, but that doesn't mean we stop trying to get points. The season is far from over," he continued.
"I'm just pleased the lads kept their nerve the way they did. People were panicking and newspapers predicting the end of the world, but the players held their nerve."