Clitsome, 28, hasn't played since Dec. 17 in Buffalo.
The native of Gloucester, Ont., has 12 points in 32 games this season. He was claimed off waivers from the Columbus Blue Jackets in February 2012 by the Jets.
The team didn't disclose the nature of the injury but said he had an operation at the Marina Spine Center in Marina del Rey, performed by back specialist Dr. Robert Watkins, Sr.
Watkins is billed as having treated more professional athletes with spinal injuries than any other surgeon in the world.