Jenkinson, who was on Arsenal's books from 2011 to 2019, was a free agent following the expiration of his contract with Forest.
"I knew within five minutes of speaking with Arthur [Papas, Jets coach] he was a manager I wanted to play for," Jenkinson said. "We have a similar outlook on the way the game should be played and I know he has big ambitions for the club moving forwards.
"I'm excited to be a part of this team and I will give my all to achieve something special here."
"Carl's signing is a [win] for the club and the region," manager Arthur Papas added. "Carl is able to fill any position across the backline, suits our attacking way of playing and will play a key role in taking us a step closer to achieving our ambitions."