"Ben could be a couple of weeks... he's got a stress reaction at the bottom of his shin," Dyche told a news conference on Thursday.
"It's a response to the workload and gets quite sore around where the ankle attaches to the shin.
"He had something similar last year which cost him a few games. We will just have to monitor that and see how it settles down," the manager added.
Versatile defender Kevin Long is expected to step in for Mee while winger Johann Berg Gudmundsson could return to the squad after missing the last two games with a calf injury.
Burnley are seventh in the league table with 49 points from 32 games while Leicester are one place and six points behind them.
(Reporting by Shrivathsa Sridhar in Bengaluru; Editing by Toby Davis)
