The Watford striker has been in superb form this season and has formed a deadly partnership with Odion Ighalo at Vicarage Road.
Deeney has been in fine goal-scoring form with six goals in 11 appearances this season.
Despite being aware of the fact that Watford are unlikely to sell any of their strikers in January, Wenger is prepared to test the water with a significant offer of around £20m.
Arsenal are in the market for a striker and Wenger is aware that adding a goalscorer could give him an edge over title rivals Manchester City.
Earlier this week Arsene Wenger has revealed that he is looking to land two players during the January transfer window and it seems Deeney is his second target.
Arsenal have already agreed on a deal to sign the Basel midfielder Mohamed Elneny. The highly rated midfielder is expected to fly in today and complete the formalities.