The Frenchman wants to leave the Stamford Bridge club in the summer.
He is unhappy at a lack of game time, and his agent is working to find him a new club.
And according to Defensa Central, an agent has offered Real Madrid the chance to sign the veteran forward.
Real loaned out Luka Jovic this window, and if he departs they could want a replacement.
However, the report indicates they are not yet interested in Giroud.