Coming out of Georgia, Jenkins was considered one of the top linebacker prospects and received an invitation to the NFL combine. He attended the combine and completed the majority of drills until he suffered a hamstring injury that made him unable to perform the short shuttle and three-cone drill. Coincidentally, teammate Leonard Floyd also injured his hamstring at the combine and was unable to perform the three-cone and short shuttle. On March 16, 2016, he opted to participate at Georgia's pro day, along with Leonard Floyd, Malcolm Mitchell, John Theus, Keith Marshall, Marshall Morgan, Sterling Bailey, Quayvon Hicks, and ten other players. Jenkins completed the short shuttle and three-cone drill, while also running positional drills and lowering his time in the 40-yard dash. He also elevated his vertical to 38" and increased his number of reps on the bench press to 19. He was projected to be a third round pick by the majority of NFL draft experts and scouts. After he completed the pre-draft process, Jenkins was ranked as the eighth best linebacker prospect in the draft by NFL analyst Mike Mayock, the ninth best outside linebacker by NFLDraftScout.com, the ninth best linebacker by ESPN, and was ranked the 11th best linebacker by Sports Illustrated.
Jenkins was drafted by the New York Jets in the third round (83rd overall) in the 2016 NFL Draft. Jenkins was the second linebacker drafted by the Jets in 2016 behind Darron Lee (first round, 20th overall) and was the ninth linebacker selected in the entire draft. Fellow Georgia linebacker and teammate, Leonard Floyd, was selected in the first round (ninth overall) by the Chicago Bears.
He suffered a calf injury in mid-August and was unable to play in the first two games of the season. On September 25, 2016, Jenkins made his professional regular season debut and first career start, recording five combined tackles in a 24–3 loss to the Kansas City Chiefs. During a Week 14 matchup against the San Francisco 49ers, he recorded four combined tackles and was credited with a half a sack on Colin Kaepernick in the Jets' 23–17 overtime victory. The next game, Jenkins collected five combined tackles and made his first career solo sack on Miami Dolphins' quarterback Matt Moore, as the Jets lost 34–13. Jenkins finished the 2016 season with 41 combined tackles (23 solo), 2.5 sacks, a forced fumble and two passes defensed in 11 starts and 14 games.