Wenger was widely tipped to leave Arsenal last summer after the Gunners failed to qualify for the Champions League, but the Frenchman ultimately penned a new two-year contract with the club.
The 68-year-old is expected to decide to step down in the summer of 2019, and according to the Daily Star, he sees Van Bronckhorst as the perfect candidate to take on the role.
Van Bronckhorst, 42, spent two seasons as a player under Wenger at Arsenal, and the pair are still said to have a strong relationship.
The former Netherlands midfielder led Feyenoord to their first league title in 18 years last season, while his career as a player has made him a respected figure in the European game.
Arsenal are currently sixth in the Premier League table ahead of their trip to Burnley this weekend.