"Andre suffered a small fracture in his left foot during a non-contact drill," said Cincinnati Coach Marvin Lewis. "Our medical staff tells me the injury will not require surgery. It looks like he's going to miss a few weeks."
Smith, the sixth overall pick in this season's draft out of Alabama, signed a deal reportedly worth up to $42 million after a 30-day holdout.
Smith is the Bengals' fifth first-round pick in six years to suffer a major injury in his rookie season.
Chris Perry played in just two games in 2004 because of a sports hernia; David Pollack suffered a knee sprain in 2005 and broke his neck in 2006; Kenny Irons tore his ACL in a pre-season game in 2007; and Keith Rivers missed nine games with a broken jaw last year.