Mays, who was expected to start alongside Derrick Johnson at inside linebacker, is scheduled to have a follow-up examination late next week.
Mays signed a two-year, $6 million contract with the Chiefs in the offseason, and he was being counted on to replace the departed Akeem Jordan. Mays had been having a productive training camp as a run-stuffing linebacker who would sub out in obvious passing situations.
The Chiefs were already dealing with a rash of injuries.
The injury to Mays should create playing time for James-Michael Johnson, who appeared in all 16 games for the Chiefs last season, and Josh Mauga, a veteran who has moved past second-year pro Nico Johnson on the depth chart after signing at the start of training camp.
Mauga has also missed time in training camp with injuries.
Information from The Associated Press is included in this report.