Slauson suffered the injury in the second half of the Chargers' shutout win against the Denver Broncos on Sunday.
"He was one of our captains, so you're going to lose some leadership there in that offensive line room," Chargers coach Anthony Lynn said. "He was a like another coach in that room, especially for the young players. So he had a lot of good intangibles. It wasn't just what he did on the field. So we're definitely going to miss Matt."
Slauson signed a two-year, $4 million deal last offseason in joining the Chargers, solidifying the center spot by playing 964 snaps in 2016.
The Chargers moved the 31-year-old, veteran to his more natural position of left guard this season, where he made seven starts and played 402 snaps on offense.
Rookie third-round selection Dan Feeney will take over at left guard for Slauson. The Indiana University product played 22 snaps at left guard, 21 snaps at right guard and five snaps as an extra tight end this season in seven games.
"Dan's been growing like a weed, and he's ready," Lynn said. "We knew that if something happened that he could step in and fill that role."
The Chargers already lost second-round selection Forrest Lamp for the year after he suffered an ACL right knee injury during training camp that required surgery. They also lost 2016 seventh-round selection offensive lineman Donovan Clark in August for a second straight year due to an ACL knee injury.
The Chargers will play their third different starting offensive line combination in eight games on Sunday when they face the New England Patriots.
Lynn said the Chargers will likely bring in another offensive lineman to add depth up front this week.