Weatherspoon, who calls the defensive signals for the Falcons, suffered the injury in the team's joint practice with the Cincinnati Bengals on Monday.
"It was an open dislocation. We are going to keep him out, and the time frame is going to be based on how quick the wound heals. It's a matter of the wound healing," Smith told the newspaper.
Weatherspoon, 25, finished second on the Falcons with 95 tackles last season. He also had three sacks and an interception.
Smith also told the newspaper that Julio Jones also will not play in Thursday night's preseason opener against the Bengals because of tightness in his hamstrings.
"It is a precautionary move," Smith told the newspaper.