The 6-foot, 225-pound Smith suffered the injury during practice Saturday at Levi's Stadium.
Smith, 28, joined the 49ers as a free agent on March 9 after signing a five-year, $26.5 million deal. He played in 15 games, including 14 starts, for the Oakland Raiders last season, and he started all 16 games in 2015, when he had four sacks for the Raiders.
Smith was the MVP of Super Bowl XLVIII following the 2013 season while playing for the Seattle Seahawks.
"He was unbelievable. Looked like the Super Bowl MVP," 49ers defensive coordinator Robert Saleh said on Monday. "I don't hurt as much for the organization as I do for him and all the hard work he put in this offseason to position himself to have the best year of his career.
"For that to happen to him when it happened to him, I feel sick for him."
The 49ers signed linebacker Austin Calitro on Monday to add depth at the position.
The 6-0, 240-pound Calitro originally signed with the New York Jets as an undrafted rookie free agent on May 5 out of Villanova and was later waived on May 15. During his time at Villanova (2012-16), he appeared in 48 games and registered 220 tackles, 26 tackles for loss, 8.5 sacks, eight passes defensed, three forced fumbles and two interceptions.
Calitro, 23, tied the team lead with 86 tackles and also registered 12 tackles for loss and 3.5 sacks in his senior season in 2016.