After being examined by Dr. Michael Watkins, Crisp is expected to be sidelined until at least the All-Star break -- which takes place July 13-16 -- according to the San Jose Mercury News.
Crisp suffered a setback to the injury that plagued him multiple times in 2014 while making a diving catch against the Houston Astros on May 19.
The 35-year-old, who missed the first month of this season, has two hits in 45 at-bats since debuting on May 6.
A surgical option is available to Crisp but on the condition that he not play baseball again. Crisp and the team are exploring alternatives that would allow him to continue playing when healthy.
"They are recommending an epidural, and he'll get that tomorrow," A's manager Bob Melvin said. "But we are looking at at least a month of no baseball activity after that."
Kazmir left Wednesday's start with shoulder pain in the third inning and will be skipped in the rotation for one start before being re-evaluated.
Assuming a five-man rotation, Kazmir's next turn would come Monday, but the A's are off until Tuesday and won't have to disturb the rotation on account of Kazmir's absence.
"We're looking at him miss just one start," Melvin said. "The MRI showed just a bit of a weak shoulder.
"There is some relief [with the news]. You could kind of tell with him yesterday that that was the case. He was open to going back out there. But we just wanted to be careful with him, with good reason."