No details are available regarding how Claiborne suffered the apparent injury.
Claiborne started 15 games for the Jets last season, and he is expected to serve in a starting role once again in 2018.
Claiborne spent the first five seasons of his career with the Dallas Cowboys after they selected him with the No. 6 overall pick in the 2012 NFL draft out of LSU.
While he showed flashes of brilliance at times when he played, Claiborne struggled to stay healthy in Dallas. Over the course of five seasons, Claiborne missed 33 games due to injury.
That resulted in Claiborne signing with the Jets and having arguably the best season of his career in 2017, as he finished with 43 tackles, eight passes defended and one interception in 15 appearances.
The 28-year-old veteran played well enough last season that New York signed him to a one-year, $7 million contract for 2018, per Spotrac.
New York also signed former Los Angeles Rams cornerback Trumaine Johnson to serve as the team's No. 1 corner.
For as long as Claiborne is on the shelf, look for Buster Skrine, Juston Burris, Darryl Roberts, Rashard Robinson and rookie sixth-round pick Parry Nickerson to be among the cornerbacks who will receive increased reps.