According to Rapoport, Clowney will have surgery Monday and miss four to six weeks.
Clowney was the No. 1 selection in April's draft. He set a South Carolina record in 2012 with 13 sacks.
Whitney Mercilus, who started all 16 games for Houston last season, is listed as Clowney's backup.
Clowney had one tackle in limited action Sunday as the Texans beat Washington, 17-6.
- Dan Gartland