The Bees are looking for £6 million for their prized winger, but the Owls have been undeterred by the high asking price.
Wednesday have money to spend in January, and appear to have targeted some of the top performing Championship players.
Carlos Carvalhal’s men have already had a bid accepted for Reading’s Nick Blackman, according to Sky Sports, and Judge is next on their wishlist.
Wednesday already have one of the largest squads in the Championship, but Carvalhal is keen to add even more quality.
Judge has scored 10 goals in 23 matches this season, and looks one of the most dangerous attacking threats in the second tier.