Couture was taken to a Nashville hospital after suffering the injury late in the second period.
Sharks coach Pete DeBoer did not have an update Saturday on Couture's status. San Jose's next game is Tuesday when the Sharks host the New York Rangers.
"You can't replace him, so it would be really tough," Patrick Marleau said of potentially losing Couture, according to CSN Bay Area. "But if that is the case, then guys are going to have to pull up the slack. Definitely we hope he's back sooner."
A shot by Sharks defenseman Brent Burns ricocheted off a stick and was sent into Couture's face. Couture, 27, appeared to lose a tooth from the impact, and he then skated to the locker room.
Couture is third on the Sharks in points this season with 52. San Jose enters Sunday tied for the Pacific Division lead with the Anaheim Ducks and Edmonton Oilers.
Information from The Associated Press contributed to this report.