Mays, who was expected to start alongside Derrick Johnson at inside linebacker, will likely have the surgery within the next 24 hours, a league source told ESPN's Adam Caplan.
The Chiefs play the Vikings in their third preseason game Saturday night.
Mays signed a $6 million, two-year contract with the Chiefs in the offseason, and he was being counted on to replace the departed Akeem Jordan. He had been having a productive training camp as a run-stuffing linebacker who would then sub out in obvious passing situations.
The Chiefs were already dealing with a rash of injuries.
Pro Bowl running back Jamaal Charles returned to practice Thursday after missing a week with a foot injury, though he is not expected to play against Minnesota. Pro Bowl safety Eric Berry (heel) and wide receiver Dwayne Bowe (quad) are among several other Chiefs players who have not been practicing.
"Listen, I don't care about all that. Next guy is in and we roll," Chiefs coach Andy Reid said this week, when asked how the injuries have affected preparation for the season. "It's simple that way. If you can play, you can play, and if you can't, you can't."
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 was used in this report.