— AC Milan (@acmilan) September 4, 2020
Diaz, 21, will spend the next season gaining experience in Serie A under Milan boss Stefano Pioli.
Milan do not have an option to buy Diaz at the end of his loan spell, but reports state that may be revisited down the line.
Brahim joined Madrid from Manchester City in January 2019, but has found game time hard to come by.
He has made 21 appearances for Los Blancos so far, scoring two goals.