Chiefs trainer Rick Burkholder announced Smith will not need surgery, but it will take six weeks for him to make a full recovery, ESPN's Adam Teicher reports.
Smith was injured during the third quarter of last week's loss to the Pittsburgh Steelers. He remained in the game and finished with 311 yards on 31 of 45 passing. In 15 games this season, Smith has completed 65.3 percent of his passes for 3,265 yards, 18 touchdowns and six interceptions.
The Chiefs are 8-7 and must beat the Chargers in order to keep their playoff chances alive in Week 17.
Backup Chase Daniel, who has attempted just one pass this season, will get the start on Sunday. He started in Week 17 of the 2013 season, completing 21 of 30 passes for 200 yards and one touchdown in an overtime loss to the Chargers.
The Chiefs host the Chargers at 1:00 p.m. ET on Sunday.
- Paul Palladino