The left-back picked up a knock to his ankle 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.
Loanees Toumani Diagouraga, Wojciech Szczesny and Tommy Smith should be added to the Brentford squad.
French midfielder Diagouraga this week arrived on an initial month-long deal from Peterborough and will probably be on the bench at Carrow Road.
Arsenal goalkeeper Szczesny will reclaim his place if the Bees can extend his loan spell in time, and Ipswich defender Smith returns after being denied permission to play in the midweek FA Cup defeat by Doncaster.
Tottenham teenager John Bostock will also be in the squad, although his current spell expires after the game.
A fifth loanee, Rhys Murphy, has gone back to Arsenal after suffering an ankle injury but fellow striker Carl Cort returns having been ineligible on Tuesday night.
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