A league source told ESPN NFL Insider Adam Schefter that Green's deal is for two years.
To make room for Green on the roster the Jets placed Jenkins on injured reserve Wednesday.
Jenkins will miss the rest of the season after tearing the anterior cruciate ligament in his surgically repaired left knee in the team's opener against Baltimore on Monday night.
The extent of the injury was determined Tuesday following an MRI exam.
Green, listed as 6-foot-2 and 365 pounds, appeared in 12 games for the Jets last season, registering 16 tackles. He signed with the Washington Redskins this offseason but was cut before the season began.
After Jenkins' injury, the Jets were down to four veteran defensive linemen: Shaun Ellis, Mike Devito, Sione Pouha and Vernon Gholston, a converted linebacker. They have Matt Kroul and Marcus Dixon on the bench. In the preseason, they lost Ropati Pitoitua (Achilles tendon) to a season-ending injury.
Jenkins tore the same ACL midway through last season and used a graft from a hamstring to repair it. Monday night's game was his first in the regular season since that injury.
Jenkins said in training camp that he used the time while he was rehabilitating to do some soul searching, and rededicated himself to football after contemplating retirement. He has often spoken about how he now considers himself a family man first and a football player second, and the latest injury may test his desire to play again.
On Tuesday, Jets coach Rex Ryan said Jenkins hadn't spoken to him about his long-term plans.
"I think he's going to get away from it, and then he'll make a decision," Ryan said.
Information from ESPNNewYork.com's Rich Cimini and The Associated Press was used in this report.