The left-back picked up the injury during last weekend's 5-0 thrashing of Colchester and is struggling to make kick-off.
Simon Lappin could move back from midfield to cover if Drury is ruled out, with Stephen Hughes and Matt Gill available to fill in in midfield.
Defender Michael Spillane is back in training after three months out with a hamstring injury but is not yet ready for a return while Jens Berthel Askou remains sidelined with his broken metatarsal.
Paul Lambert's side can go top of League One this weekend with a win as Leeds are in FA Cup action against Tottenham.
