Boro will play the first leg away on Wednesday/Thursday 15th/16th February and the second leg at home on Thursday 23rd February.
If successful against Stuttgart, Boro will play Roma or Brugge in the Round of 16.
Round of 32 draw:
Litex Lovech v Strasbourg
Stuttgart v BORO
Slavia Prague v Palermo
Heerenveen v Steaua Bucharest
Lokomotiv Moscow v Sevilla
Bolton v Marseille
Hertha Berlin v Rapid Bucharest
Basel v Monaco
Udinese v Lens
Rosenborg v Zinit St Petersburg
Brugge v Roma
Schalke v Espanyol
Lille v Shakhtar Donetsk
Thun v Hamburg
Real Betis v AZ Alkmaar
Artmedia Bratislava v Levski Sofia
Tie to be played 15th/16th February and 23rd February.
Round of 16 draw:
Hertha Berlin/Rapid Bucharest v Thun/Hamburg
Basel/Monaco v Litex Lovech/Strasbourg
Stuttgart/BORO v Brugge/Roma
Heerenveen/Steaua Bucharest v Real Betis/AZ Alkmaar
Slavia Prague/Palermo v Schalke/Espanyol
Bolton/Marseille v Rosenborg/Zinit St Petersburg
Udinese/Lens v Artmedia Bratislava/Levski Sofia
Lille/Shakhtar Donetsk v Lokomotiv Moscow/Sevilla
Ties to be played 9th and 15th/16th March.