Although a timetable for his return is unclear, there's hope that his recovery time will be shorter than three months, the source said.
Foster suffered the injury during the Texans' first fully padded practice of training camp Monday night in Houston. The tear came during a pass play, and Foster did not return to practice.
While the injury initially didn't raise alarms, a source said a subsequent MRI revealed it was more serious.
Foster will miss several regular-season games, and it's possible he will be placed on injured reserve with a designation to return. The earliest the Texans can make that move is Sept. 6 at 4 p.m. ET.
"We didn't make a move immediately because we felt like we were OK," Texans general manager Rick Smith said. "Chris Polk is on the mend and so hopefully he'll be back soon. We'll just keep adjusting to whatever happens out here on the field."