The Jets announced the move Thursday night, a few hours after bringing Johnson in for a workout. To make room on the roster, New York waived undrafted rookie quarterback Jake Heaps.
Ryan Fitzpatrick is the starting quarterback with Geno Smith sidelined several more weeks with a broken jaw. Fourth-round draft pick Bryce Petty is currently Fitzpatrick's backup. The Jets also signed veteran Matt Flynn last week, but he has yet to fully practice with a sore hamstring.
Johnson was a fifth-round pick of Tampa Bay in 2008, and also has been with Cleveland and had two stints each with Cincinnati and San Francisco. The Bengals released him Tuesday after A.J. McCarron won the backup job.
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