Bouman has appeared in all 13 games for the Saints this season but mainly on special teams as a holder. He did play at quarterback in one game this season in early October against the Green Bay Packers. He completed 5-of-13 passes for 31 yards and an interception.
Haslett said the move was made because of the team's 3-10 record.
"It's not an indictment of Aaron, it's an indictment on our record and where we're at," Haslett said. "It's an opportunity to see what Todd can do with the situation."
Brooks has struggled this year, throwing for 2,882 yards on 240-of-431 passing. The Virginia product has thrown 13 touchdowns, but has also been intercepted 17 times.