Crawford suffered the injury on Friday after he beat out an infield hit. He will be eligible to be activated on July 3, at which point the Red Sox will be playing the finale of a three-game series in Crawford's hometown of Houston.
"That's what we're shooting for -- as soon as possible," Crawford said. "We're hoping for a speedy recovery. I definitely don't want to miss that [whole] series. We'll be cutting it close, that's for sure. Hopefully, things will get better soon."
In Crawford's place, Josh Reddick was promoted from Triple-A Pawtucket for the second time this season. With lefty Randy Wolf starting for the Brewers on Saturday, Reddick did not get the start. Instead, it was Darnell McDonald, who played left. Mike Cameron subbed for J.D. Drew in right.
"We'll probably fill in with Mac, Cam, Josh, maybe a little [Drew] Sutton. Kind of piece it together a little bit," said Francona.
Reddick was looking forward to his latest stint in the big leagues and might start Sunday against righty Yovani Gallardo.
"Hopefully, I can get in there against righties, play some left field,," said Reddick. "I hadn't done it a whole lot this year. I've done it in the past so I feel pretty comfortable out there."
Much like right-hander Michael Bowden, who is currently in Boston's bullpen, Reddick has spent a lot of time on the Boston-Pawtucket shuttle the last few years.
"Yeah, I told him last year, or the beginning of this year, they're going to rename that Lou Merloni highway after me or him," said Reddick. "Hopefully, we can keep that trend. Rather be that guy than the guy who doesn't get the call at all. You have to look at it as a positive thing."
As Reddick matures, he thinks he's gotten far better at staying in his routine, whether it's in Boston or Pawtucket.
"I found real fast that you have to stay in your own routine, whether you're playing or not playing, and last year, that got really tough and frustrating," Reddick said. "I just didn't stick with it. Now this year I've kind of stuck with my routine, whether I was playing down there or up here. It seems to be keeping me in line a little more, keeps me in my groove a little bit better."
This was Crawford's second career trip to the disabled list, and first since 2008, when he suffered a subluxation of his right middle finger.
"With a hamstring, you don't want to mess around with that," Crawford said. "You see a lot of guys try to come back too early, and they mess it up, and they end up being out even longer. We just want to make sure when I come back, I stay there."