He will stay with Pawtucket until he is ready to be in Boston's lineup.
"Seven to nine innings at shortstop tomorrow, followed by a DH, then shortstop again," said manager John Farrell of the plan for Drew. "In talking with him earlier today, he's starting to feel like he's getting his legs underneath him. We can't skip any steps along the way to do that. He's progressed as we had hoped."
The Red Sox re-signed Drew on May 21.
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.