Mee was forced off in the second half of Monday’s 4-2 defeat at the London Stadium and scans have shown the centre-back faces a significant spell out.
Losing Mee is a huge blow to Brentford in their battle against relegation.
The Bees are just five points above the bottom three and are in danger of being dragged further into trouble as their injury crisis deepens.
Brentford are already without Ethan Pinnock until after the March international break and Thomas Frank will need club-record £23million signing Nathan Collins to step up after an indifferent season.
Mee has been playing on the left on a three-man defence for Brentford since Frank switched formation last month.
Collings replaced him against West Ham, having been dropped earlier this month following some poor performances.
Brentford play Chelsea at home on Saturday and also face Arsenal, Manchester United and Brighton in their next five fixtures.
