Stefanski said he was "sick'' over Conklin's injury.
"Obviously, Jack's a big part of what we do,'' Stefanski said Monday. "He's a good man, good teammate.''
Stefanski wouldn't speculate whether Conklin would be back in time for training camp next year.
Conklin went down on the Browns' second offensive possession. After being helped to the sideline by the Browns' training staff and his teammates, he was carted to the locker room.
Conklin had just returned from a dislocated elbow that he suffered in Week 8.
He was an All-Pro selection in his first season with Cleveland in 2020 after signing a three-year, $42 million contract that included $30 million guaranteed.
ESPN's Jake Trotter and The Associated Press contributed to this report.