The move comes after tight end Chris Herndon was handed a four-game suspension by the NFL for violating the NFL's substance abuse policy, stemming from a June 2018 drunken-driving arrest in New Jersey.
Griffin became a free agent when the Houston Texans released him in May, just weeks after he was arrested on charges of vandalism and public intoxication for allegedly punching out a front window in a hotel in Nashville, Tennessee, which was hosting the NFL draft.
Police in Nashville say hotel security followed Griffin after the window was broken, and when officers responded to the call, Griffin's left hand and fingers were bleeding and he appeared intoxicated and unsteady on his feet.
Police say Griffin was arrested for his own safety rather than given misdemeanor citations. A friend of Griffin's said he had been overserved, according to multiple reports.
Charges against Griffin were ultimately dismissed.
News of Griffin's deal with the Jets was first reported by NFL Network.
Griffin, who turned 29 in January, had 24 receptions for 305 yards last season and has 136 career receptions and seven touchdowns in six seasons, all with the Texans, who selected him in the sixth round (201st overall) of the 2013 draft.
In other roster moves, the Jets placed free safety Marcus Maye on the active/physically unable to perform list. This means Maye, who had shoulder surgery following last season and was limited throughout the offseason, can be activated at any time.
Rookie cornerback Blessuan Austin, a sixth-round pick out of Rutgers, was placed on the active/non-football injury list. He missed most of last season after major knee surgery.
ESPN's Rich Cimini and The Associated Press contributed to this report.