Cooper Rush
The Bengals are familiar with the final two guys on our list and not for good reasons. The first entry, Cooper Rush, upset the Bengals in Week 2 of this past season when he and the Cowboys managed to win a close game to send Cincy to 0-2 on the year.
Rush played well while replacing an injured Dak Prescott in Dallas and led the team to four wins and one loss. He appeared in nine games total, throwing for 1,051 yards, five touchdowns, and three picks. If the Bengals want to sign a proven backup, Rush is one of the better options.
Mike White
Another guy the Bengals are familiar with for all of the wrong reasons is Mike White, who was the Jets backup in 2021 and throttled Cincinnati in an upset loss. It's a game that Bengals fans want to forget but as long as White is in the league, it'd be hard to do that.
White is probably one of the better backup quarterbacks available in free agency and for good reason. He looked good when starting for the Jets in 2022, tossing 1,192 yards, three touchdowns, and four interceptions. His best game was against the Bears where he threw for 315 yards and three touchdowns.
I wouldn't be shocked if the Jets brought White back but if they don't and the Bengals need a backup quarterback, Mike White would be a good choice.