Culpepper auditioned for the team earlier this week, but the team decided not to sign him.
McCown is a former Arizona Cardinals third-round draft pick who played collegiately at Sam Houston State.
The older brother of NFL quarterback Luke McCown, Josh McCown has also played for the Detroit Lions, Oakland Raiders and the Miami Dolphins as well as a stint with the Hartford Colonials in the United Football League.
McCown has passed for 6,582 career yards, 35 touchdowns and 40 interceptions.
McCown has also rushed for 569 yards and three touchdowns