To get him down to the Minor Leagues, the Red Sox first had to get him through waivers, which Drew cleared on Friday at noon ET. Once he cleared waivers, he was optioned to Greenville.
Drew will play for Greenville for four days, playing twice at short, once at designated hitter and probably getting a day off. On Friday, he played five innings, going 1-for-2 with a single and a strikeout.
He will then move up to Double-A Portland or Triple-A Pawtucket.
Drew needs to stay in the Minor Leagues for 10 days before he can be recalled by the Red Sox, due to option rules.
The earliest Drew will debut for Boston is June 2, when the Red Sox open a three-game series in Cleveland.
"He'll go seven innings probably two or three times," said Red Sox manager John Farrell. "We have a minimum of 10 days to execute the plan that's laid out with the target of 20 to 25 at-bats or so to be had. That's where we are."
With Drew vacating a roster spot, the Red Sox recalled right-hander Alex Wilson from Triple-A Pawtucket for the second time this season.
Wilson will fill a spot in the bullpen for the next two days and probably be sent down on Sunday, when Boston needs a starting pitcher to replace Felix Doubront, who is on the disabled list.
Farrell reiterated that Brandon Workman and Allen Webster are the top two candidates to make that start.
Ian Browne is a reporter for MLB.com. Read his blog, Brownie Points, and follow him on Twitter @IanMBrowne. This story was not subject to the approval of Major League Baseball or its clubs.