Crawford played catch at 110 feet before Saturday's game.
"He threw the ball really well," manager Bobby Valentine said. "He's going to take batting practice in a little while. Yesterday he hit it really well. I've got to think he's getting pretty close to playing in a baseball game."
No date has been mapped out for Crawford's Minor League rehab assignment. He hasn't played a game for the Red Sox this season.
Crawford was initially coming back from left wrist surgery, but he then suffered a strained UCL in his left elbow.
Center fielder Jacoby Ellsbury is currently training in Fort Myers, Fla., and could also be relatively close to a Minor League rehab assignment.
Ellsbury has been out since April 13, when he suffered a subluxation (dislocation) of his right shoulder.