The Jets released cornerback Darrelle Revis last week and need secondary help after they struggled in pass coverage last season.
Claiborne, who turned 27 in February, was off to the best start of his career in 2016 after re-signing with the team on a one-year deal, but he suffered a groin injury on Oct. 30 that sidelined him for nine games. He returned for the playoffs and suffered a hip injury in a loss to Green Bay.
For the season, he finished with 26 tackles, an interception, a fumble return and five pass deflections in seven games.
Injuries have plagued Claiborne's career since the Cowboys traded up to get him with the sixth overall pick in the 2012 draft.
He had wrist surgery before the draft that limited his offseason work as a rookie, but he managed to start 15 games and finished with 55 tackles and one interception. From 2013 to '16, he managed to play more than 10 games in a season just once (11 in 2015). He tore his patellar tendon in 2014 and missed the final 12 games of the season. He made a solid recovery in 2015 but missed five games with hamstring and ankle injuries.
For his career, he has just four interceptions and has never had more than one in a season. He also has four fumble recoveries and 151 tackles.
The Jets also re-signed defensive tackle Mike Pennel, his agent announced on Twitter. They claimed him on waivers in January, but declined to tender him as a restricted free agent. They brought him back on a one-year contract.
Pennel (6-foot-5, 360 pounds) spent his first three seasons with Green Bay, showing promise at times, but he was suspended twice last season for violating the league's substance abuse policy. He will provide interior depth along the Jets' defensive line.
ESPN's Todd Archer and Rich Cimini contributed to this report.