Stefanski didn't provide specifics of the injury, but a source told ESPN's Adam Schefter that tests revealed that Conklin tore the ACL and MCL in his left knee.
Conklin will undergo season-ending surgery.
"Very, very disappointed for Jack," Stefanski said Monday. "He's a huge part of this team. When he gets this surgery, and when he's ready, he's going to be back in this building helping us, because he's a big part of that offensive line room."
Conklin injured his left knee while pass protecting in the first half against the Bengals. Defensive end Trey Hendrickson rolled into Conklin's knee while rushing from the other side of the Browns' pocket.
Conklin was an All-Pro his first year in Cleveland in 2020 but has battled injuries since, including a torn patella in his right knee in 2021.
Rookie Dawand Jones, a fourth-round pick out of Ohio State, replaced Conklin in Sunday's game.
"Some really good moments, some things that he's got to clean up," Stefanski said. "Honestly, any young player, when you get thrown into the mix early, there's going to be some time to catch up. And I thought he had some good moments, some things that we will work real hard to clean up."
The Associated Press contributed to this report.