Koppen was injured late in the first half of Monday night's 38-24 win against the Dolphins and was carted off the field.
He underwent an MRI on Tuesday, a source told ESPNBoston.com's Mike Reiss.
"That's tough. He's the best and there's no one like him," Patriots quarterback Tom Brady said on Tuesday. "I think he and I have a great chemistry between the two of us. We always have and he's one of my great friends. Hopefully he's back at some point. "
Dan Connolly took Koppen's place at center. The Patriots already are thin on the offensive line with starting right tackle Sebastian Vollmer sidelined by a back injury. Rookie Nate Solder made his pro debut in his place.
Vollmer returned to the field on Wednesday.
On Wednesday, the Patriots also signed offensive lineman Donald Thomas.
Thomas was drafted by Miami in 2008 in the sixth round out of Connecticut and spent two seasons with the Dolphins and one with Detroit. He signed with the Lions last Nov. 23 and was inactive for the last six games. He was released on Sept. 3.
In other moves Wednesday, the Patriots released offensive lineman Thomas Welch, signed tight end Garrett Mills and linebacker Jeff Tarpinian to the practice squad and released linebacker Aaron Lavarias from the practice squad.
Information from ESPNBoston.com's Chris Forsberg and The Associated Press was used in this report.