It seems doubtful because there is no real financial (or talent) incentive that would make Carr much of an improvement over Ryan Tannehill.

Source: https://titansized.com/2023/01/16/tennessee-titans-qb-carousel/

Not sure why Woods went there in the first place, knowing Ryan Tannehill was going to be his QB.

Source: https://profootballtalk.nbcsports.com/2023/02/22/titans-release-robert-woods/

Quarterback Ryan Tannehill isn’t as shifty as Tagovailoa and doesn’t have as many speedy targets, but he can be an effective runner if pressured into a scramble out of the pocket.

Source: https://www.dailynews.com/2023/09/16/chargers-facing-different-set-of-challenges-in-week-2-vs-tennessee/

Ryan Tannehill has over a 100 passer rating in two of his last three games, contributing to their three-game winning streak.

Source: https://www.nationalfootballpost.com/best-bets/anytime-td-scorer-props-picks-longshots/

The Jets could also look at other quarterback options such as Lamar Jackson, Jimmy Garoppolo and Ryan Tannehill.

Source: https://www.bostonherald.com/2023/02/28/jets-had-a-fantastic-visit-with-qb-derek-carr-gm-joe-douglas-says/