The shortstop is expected to play for the Triple-A Pawtucket Red Sox on Wednesday and Thursday, when they host the Gwinnett Braves.
Drew is coming off a three-day stay with the Single-A Greenville Drive, where he went 3-for-8 with two doubles, two runs batted in and one walk. He signed with the Red Sox on May 21 and was optioned to Greenville on Friday.
The club will have to make a corresponding roster move, according to the PawSox, because Drew is not on a rehab assignment.