Street was placed on the 15-day disabled list on May 5 with a strained right lat muscle.
Manager Bud Black says Street is likely to throw a long-toss session sometime soon. A bullpen session could follow. The results of the bullpen work will determine the next move.
"He feels good," Black said. "He went free and easy, no pain."
Street, who was 4-for-4 in save opportunities before the injury, is expected to make at least two Minor League rehab appearances before resuming his closer duties.