Klein was injured while blocking a shot during the Rangers' win over the Washington Capitals on Wednesday. He then returned to New York to have his left arm examined by a team physician and underwent X-rays.
According to the New York Post, the Rangers will not be able to recall a defenseman in the immediate future to due salary cap constraints.
The 30-year-old Klein has scored a career-high nine goals and has tied a career-high with 17 assists this season. In 65 games, he has a plus-24 mark and averages 18:29 time on ice.
The Rangers acquired Klein last season in a trade with the Nashville Predators.
New York (42-17-7) plays the Sabres in Buffalo on Saturday.
- Molly Geary