"He's week to week but he'll probably be gone for the next couple weeks," Lynn said on SiriusXM NFL Radio on Thursday.
Bosa initially injured his foot during 11-on-11 team drills in practice on Aug. 7. The Pro Bowl standout sustained the current injury during practice and had been in a walking boot this week.
Bosa's absence means more is expected of second-year pro Isaac Rochell. The 6-foot-4, 280-pound Rochell started all four preseason games for the Chargers in place of Bosa and had four tackles against the Chiefs.
Bosa was selected to the Pro Bowl in 2017 after racking up a team-leading 12.5 sacks while starting all 16 games. The 6-5, 280-pounder finished with 70 tackles and also had four forced fumbles last season.
A first-round pick out of Ohio State (No. 3 overall) in the 2016 NFL Draft, Bosa enjoyed a stellar first season with 10.5 sacks in 12 games, receiving NFL Defensive Rookie of the Year honors.