Street, who was competing to be the Angels' closer this season, suffered a "Grade-1 plus" strain of his lat muscle during spring training and has been out since. Angels GM Billy Eppler told the Orange County Register that Street's rehab was slowed by a "mild shoulder impingement."
A two-time All-Star selection, Street, 33, went 3-2 with a career-worst 6.45 ERA and nine saves during an injury-plagued 2016 season with the Angels.
Andrew Bailey, who also was battling for the closer job this spring, also has a shoulder injury and has not thrown since going on the DL last week.
Cam Bedrosian is the team's current closer, having saved two games this season.
By moving Street to the 60-day DL, the Angels brought up reliever Parker Bridwell, who was acquired from the Orioles on Monday.