While Liverpool are now regarded as favourites to land the Germany international, Real are convinced the presence of new coach Xabi Alonso can force a rethink from the player.
Wirtz and Xabi are close after working together for three years at Bayer and the prospect of the attacking midfielder following his coach to Madrid remains a prospect, says AS.
Bayer are demanding €150m to sell Wirtz - a valuation which has already seen Manchester City drop out of the running.
Real Madrid are prepared to get close to Bayer's price, knowing Wirtz would be open to continuing to work with Xabi.
Under the Spaniard at Bayer, Wirtz exploded last season with 18 goals and 19 assists in 49 games. He also finished in 12th place in the last edition of the Ballon d'Or.