Luck underwent surgery to a finger on his right, throwing hand for an injury he sustained during a 45-38 victory over Notre Dame, Stanford's last regular-season game.
"He's going to be out of practice at least two to three weeks," Stanford coach Jim Harbaugh told ESPN.com's Ted Miller on Monday. "He's not ruled out of the game, but based on what we know right now, he won't start.
"We'll have to see how his finger heals and see what his availability is for the game. It will be a game-week decision," Harbaugh added.
The Cardinal have been anchored by running back Toby Gerhart, whose 1,736 rushing yards have put him firmly in the hunt for the Heisman Trophy.
Luck started all 12 of Stanford's games this season, leading it to an 8-4 record overall and a 6-3 mark in the Pac-10. The Cardinal finished fourth in the conference standings.
Senior Tavita Pritchard would start in Luck's place.
"I think they'll rally around him like they always have," Harbaugh said of Pritchard, who has started 19 games in his career, highlighted by a 24-23 victory over then-No. 2 USC in 2007, when the Trojans were 41-point favorites.
"Tavita has the complete respect and admiration of this team," Harbaugh said.
Harbaugh said the team was still confident of its chances for the El Paso, Texas, bowl game.
"This team doesn't make excuses," Harbaugh told ESPN.com. "There's injuries in football."