The Bosnian, missed the 5-1 Boxing Day defeat to Newcastle after injuring himself in training on Christmas Day.
“Asmir has a bad break of his finger, he is possibly looking at five to six weeks out,” said Stoke manager Mark Hughes after the game.
Begovic had played in all 17 Premier League games this season, prior to the trip to Newcastle.
He has also scored in this campaign, with his long kick bouncing over Southampton keeper Artur Boruc in a 1-1 draw in November.
Thomas Sorensen replaced Begovic at St James’ Park.