The move reunites Alexander with Jets head coach Robert Saleh, who was his defensive coordinator with the San Francisco 49ers in the 2019 and '20 seasons.
The Jets likely will use Alexander as an outside linebacker, capable of playing either the strongside or weakside positions. Only two of the three starting positions are set -- C.J. Mosley (middle) and Quincy Williams. The third spot is up for grabs, with Marcell Harris and Hamsah Nasirildeen the leading contenders.
The Jets ignored the linebacker position in free agency and the draft, leaving themselves vulnerable. Their goal was to add a veteran to the mix, which is why they visited with Alexander in May. Even though he left without a deal, they always maintained an interest.
Alexander becomes the fourth defensive player who played under Saleh in San Francisco to join the Jets since the end of the season. The others are Harris, cornerback D.J. Reed and defensive tackle Solomon Thomas.
Alexander, who suffered a torn Achilles tendon in Week 16 of the 2020 season after the New Orleans Saints acquired the linebacker in a midseason trade with the 49ers, recovered in impressive fashion last season, playing 12 games with 50 tackles, a career-high 3.5 sacks, an interception and a forced fumble.
Alexander, 27, started only eight games for the Saints because of an ankle injury early in the season, but six of those starts came in the final eight games, with him playing about 80% of the defensive snaps over that stretch.
Injuries also marred Alexander's time in San Francisco. He suffered a torn pectoral in 2019 and a high ankle sprain in 2020 before being traded to the Saints. He has good range and has been especially helpful in pass defense, with 521 career tackles, 12 sacks, eight interceptions and 10 forced fumbles.
The 6-foot-1, 227-pounder began his career as a 2015 fourth-round draft pick with the Tampa Bay Buccaneers, with whom he made his only Pro Bowl in 2017.
ESPN's Rich Cimini and Mike Triplett contributed to this report.