Coach Kliff Kingsbury previously had declined to name a starter during bowl preparations.
"We'll let them trot out there, and y'all will see [the starter] when everybody else sees," Kingsbury said earlier. "We jumped back to the basics and did a reinstall after the bowl announcement, and they have looked good. I'm excited to see 'em both play on Monday."
Brewer was the favorite to start after spring practices before a preseason back injury handcuffed his 2013 campaign. The sophomore returned late in the season and had a solid showing in mop-up duty during the season finale against Texas.
Although Brewer has been in the program for three seasons, Webb holds the clear game experience advantage. He played in nine games, starting five, during his first collegiate season. Webb was 198-of-320 for 2,315 yards, 16 touchdowns and nine interceptions.
Walk-on true freshman Baker Mayfield began the season as the Red Raiders' starting quarterback but is no longer with the team after he announced he planned to transfer. In eight games, Mayfield threw for 2,315 yards, 12 touchdowns and nine interceptions.
Information from ESPN.com's Brandon Chatmon was used in this report.