Street says his rehabilitation is "on schedule" and he is anxiously waiting the next step.
"It's progressing nicely and I'm ready to start pushing it," Street said.
Street will stretch out and go through some long-toss exercises later in the week. He hopes to graduate to a bullpen session early next week. He is with the team on its current 10-game road trip.
Padres manager Bud Black said that Street will likely need a pair of Minor League rehab outings before he can rejoin the team.
"He's getting closer to the mound," Black said.
Street, who was injured in the tenth inning of a 9-8 loss to Miami on May 4, is taking a cautious, yet optimistic, approach to his rehab.
"It's a tough muscle and you don't want to try and do too much," Street said. "You've got to be patient and do the right things. Three or four extra days can mean the difference in three or four weeks."
Street recorded four saves in his first four opportunities this season, before he was placed on the 15-day disabled list on May 5.
Street said the most difficult part of the process is the helpless feeling he has watching his team in action.
"That's the worst part," he noted. "This is baseball season. You want to play. It's not normal not to be playing."