The freak injury, described by the team as a non-displaced fracture, occurred during a center exchange. Bulger will not wear a cast while the injury heals.
"We'll see where we are after that," Rams coach Steve Spagnuolo said. "He won't do very much for two weeks, just to be on the safe side."
Kyle Boller becomes the Rams' starter while Bulger recovers. Bulger, 32, did not miss a game to injury last season or in 2006. He missed four games in 2007.
Boller, signed in free agency from the Baltimore Ravens, started eight games last season, completing 61.1 percent of his passes with nine touchdowns and 10 interceptions. Boller owns a 20-22 regular-season record as a starter, all with the Ravens. His career passer rating is 71.9.
"It'll be an opportunity for me to get out there with the ones," Boller said. "That's what they brought me in here for, in case something like this happens."
Bulger was 36-24 as a starter before losing all but four of 27 starts over the last two seasons. The Rams benched him for one game last season.
The Rams' passing game was already operating at a deficiency before Bulger's injury. Top receiver Donnie Avery could be sidelined into the regular season after suffering a broken bone in his left foot. Receivers Brooks Foster and Tim Carter suffered less serious injuries during the Rams' exhibition opener against the New York Jets.
Mike Sando covers the NFC West for ESPN.com. Information from The Associated Press was used in this report.