Begovic suffered the injury on Christmas Day and was forced to miss Boxing Day's 5-1 defeat away by Newcastle United.
The Bosnia-Herzegovina international is now likely to be out of action until February after Stoke manager Mark Hughes confirmed the extent of the injury.
"Asmir could be out for five to six weeks," Hughes said after the Newcastle game. "He did it right at the end of training yesterday and it's quite a bad break.
"We hope he's back sooner than predicted but we're going to be without him for a while."
Thomas Sorensen was called into the Stoke side at Newcastle for the veteran Danish keeper's first Premier League appearance of the season.
The injury to Begovic also prompted Hughes to recall Jack Butland a day early from his loan spell at Barnsley and he was on the bench at St James' Park.
Begovic took to Twitter to bemoan his injury and wrote: "Well not an ideal couple of days, broken finger and injustice for the boys in Newcastle! Cheers Santa!!"
However, Sorensen offered his support to his team-mate by tweeting: "I wish @asmir1 a speedy recovery as I never like seeing something bad happen to a friend and fellow pro."
Sorensen added: "Tough day to make my PL comeback. We were on top before the 2 red cards which changed the game. Got to put it behind us and move on."