"We'll see how his injury progresses during the course of the week," Shaw said. "Should hopefully make that decision by Thursday or Friday. It shouldn't be a game-day decision. We should know before game day."
Costello suffered the injury when he received a forearm to the head while in a sliding motion late in the first half. The play did not a draw a targeting penalty from the Big Ten officiating crew, something that drew Shaw's ire after he saw the replay following the game.
"I believe the officials on the field and replay officials missed the call," Shaw said. "I thought it was an ejectable offense. I've communicated with a lot of different people in a lot of different places. It was unfortunate it was missed."
The Cardinal will also be without potential All-American left tackle Walker Little, who suffered a lower body injury against Northwestern. Walker's prognosis is not clear, but Shaw said he is hopeful Little will be able to return around midseason. True freshman Walter Rouse will start in place of Walker against USC.
If Costello can't play, redshirt sophomore Davis Mills will start. Mills completed 7 of 14 passes for 81 yards in relief of Costello on Saturday.
Before he exited, Costello, a second-team All-Pac-12 selection in 2018, completed 16 of 20 passes for 152 yards with a touchdown and no interceptions.