The 33-year-old left Monday's game against the Devil Rays after five innings with cramps in both calves.
Damon reported for the Yankees' rained-out game against Tampa Bay on Wednesday and said that the right calf had not improved, stating that it "hurts like hell" and that he could "barely walk."
"I'm feeling better today than I was yesterday," Damon said on Thursday. "I'm going to get after it with the trainers the rest of the day and see where we are tomorrow. I still have sexy calves."
Manager Joe Torre said that an assignment to the DL is a possibility if Damon does not show improvement, though the team has not yet reached that decision.
Torre said that he wanted to speak with Damon before broaching the subject, as the outfielder would be the best barometer in gauging his readiness.
"You certainly don't want to push somebody and have him feel an obligation to get back if he's not physically able to get back," Torre said. "I think there are a lot of things to consider."
Damon is no stranger to playing with pain. He gritted through much of last season -- his first with the Yankees -- guarding a sore throwing shoulder and a broken sesamoid bone in his right foot.
"He's been around long enough to know about certain things he can go on with, and other things that are going to hinder what he does," Torre said. "His speed and his ability to run in the outfield, when you're dealing with leg muscles, [is something] you certainly have to take into consideration."
Melky Cabrera started in center field for the Yankees on Thursday, batting ninth. Robinson Cano assumed Damon's normal leadoff spot in the lineup, an arrangement that Torre said could continue in Damon's absence.
Damon batted .285 with 24 home runs and 80 RBIs in 149 games for New York last season. He was 1-for-2 with a run scored before being removed from Monday's game.
Bryan Hoch is a reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.