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.
The Texans have not yet added another running back to a roster that includes Alfred Blue, Jonathan Grimes, Kenny Hilliard and fullback Jay Prosch.
"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."
Timetable for Arian Foster's return
ESPN Texans reporter Tania Ganguli discusses how long RB Arian Foster may be out after having groin surgery and how Houston will fill the void while Foster is out.
How long will Texans be without Arian Foster?
ESPN analyst Field Yates goes over the possible length of the time Arian Foster could miss for the Texans with a groin injury and what the team can do as a backup plan.