Granderson hasn't played since May 24 and has played in just eight games all year after returning from a broken right forearm.
"We're getting close. I started hitting in the cage recently and doing some stuff soft toss," Granderson said, via ESPN.com. "I have been throwing. I've been running. I've been doing all my defense. Just got to get the baseball stuff underneath me in terms of hitting, getting on the field and playing and then get back up here to York."
Granderson said he has begun swinging the bat for the past couple of days. The Yankees, who have a run differential of minus two runs this season, sorely miss his power in the lineup. The Yankees come into the All-Star Break with 88 homers and a .376 slugging percentage, both ranking the team 13th in the 15-team American League.
"We'll take him when he's ready," Yankees general manager Brian Cashman said. "If pressed about when, I'd probably say early August. If sooner than that, that would be a gift from above."