He suffered the injury while throwing during the 10th inning of Friday's 9-8 loss to the Marlins at Petco Park.
In a corresponding move, the Padres selected right-handed reliever Miles Mikolas from Double-A San Antonio. To make room for Mikolas on the 40-man roster, San Diego moved infielder Logan Forsythe to the 60-day DL. The Padres originally placed him on the 15-day DL on March 26 with a fractured left foot. He began playing games at extended spring training earlier this week.
Padres manager Bud Black said there is no timetable for Street's return, and he'll likely be on the disabled list more than 15 days.
"Any time you strain a muscle, whether it's your hammy, your calf, those are muscle tears," Black said. "They heal, but they take time, so it just depends on the severity of the strain. And this is a strain that is significant enough to go on the DL and rest and there's rehab.
"I'm going to venture to say he won't pitch the day he comes off the DL, so it'll be more than 15 days. It's not one of those where he'll be ready in 15 days."
The Padres will use several pitchers to close games. Black mentioned Andrew Cashner, Dale Thayer and Luke Gregerson. The Hard-throwing Cashner is 2-1 with a 2.08 ERA. Thayer has a 0.00 ERA in three innings. Gregerson has a 3.00 ERA in 12 innings.
"I think it depends on the night, where we were the night before and usage," Black said.
Street underwent an MRI on Saturday. He pitched two-thirds of an inning in Friday's game. He exited after issuing a walk to Logan Morrison in the 10th. Street struck out Omar Infante and got Giancarlo Stanton to ground out before the walk.
Street has allowed a run in 9 2/3 innings, posting a 0.93 ERA and converting each of his four save opportunities.
"I don't think it is major as far as it isn't a shoulder or something that will require surgery," Street said.
"I've done this enough now to just take it day by day," Street added. "When you go on the DL, you know you have 15 days. We'll reevaluate in 15 days. I wish it could be quicker."