Martin exited Sunday's 40-7 loss against the Arizona Cardinals in the second quarter and did not return. He had seven carries for 23 yards.
The 27-year-old two-time Pro Bowler suffered the injury when he made a cut and planted his foot. He had an MRI on Tuesday.
Last season, the 5-foot-9, 223-pound Martin was second in the NFL with 1,402 rushing yards on 288 carries.
"When you take a top-level player out, you usually don't have exactly the same caliber player behind him, even though we're very high on Charles Sims," Bucs coach Dirk Koetter said Monday. "But yeah, it definitely hurts you. But that's the NFL. You're only healthy for one week and then you're hoping that you are fortunate enough to stay healthy the rest of the year. We didn't have good fortune in that department (against the Cardinals), but this will be a new week."
The Bucs host the Los Angeles Rams on Sunday.