Cunningham, 25, most recently spent time with the San Francisco 49ers. He was released Oct. 29 after a month with the team. He didn't make the Patriots' roster coming out of the preseason.
Cunningham, a Bronx native who was drafted in 2010, has appeared in 36 games. He has 59 tackles and 3.5 sacks. The Jets had a roster opening after placing tight end Konrad Reuland on injured reserve.