Cashman declined to state what the exact surgical procedure would be, saying that the 23-year-old Kaprielian has requested some time to weigh his options.
"Clearly, you don't bounce around the country if the surgical option isn't on the table," Cashman said. "We pointed out, you can have the surgical option, or you can try to continue to pitch through it. He's basically going to have a day or two to talk to his father and his family and his agency and then get back to us with what he would like to see happen."
Kaprielian is New York's No. 6 prospect and the No. 55 overall prospect in baseball, according to MLBPipeline.com. The club's first selection in the 2015 Draft, Kaprielian had been slated to begin the season with Class A Tampa.
Cashman said that the injury is a recurrence of the right flexor tendon strain that limited Kaprielian to just three starts last year. Kaprielian has made just 14 professional appearances since being selected 16th overall two years ago.
"Ultimately, at this point, the rehab process hasn't been successful without having a continued issue," Cashman said. "He's facing some decisions."
Kaprielian pitched to a 4.33 ERA in seven Arizona Fall League starts last year and fired two scoreless innings in Major League camp this spring, striking out three, prompting speculation that the Yankees could fast-track him to the big leagues if healthy.
"You see the ability that he possesses, but you have to be healthy to be able to harness it," Cashman said. "You know it's there, and at some point we're looking forward to having it for an extended period of time.
"Injuries are also part of the game, and you have to deal with them. Unfortunately right now, the front of his pro career has had some injuries with that elbow, but he has an opportunity to have a very long career regardless."
Bryan Hoch has covered the Yankees for MLB.com since 2007. Follow him on Twitter @bryanhoch and on Facebook. This story was not subject to the approval of Major League Baseball or its clubs.