Thiago scored seven times in 32 appearances for Bayern during the 2017-18 campaign. The 27-year-old only started 12 Bundesliga matches last term, however, and his form was indifferent for long spells.
Barcelona have been credited with an interest in their former midfielder, but according to Marca, Madrid are also considering a move due to the uncertainty surrounding the future of Mateo Kovacic.
The report claims that new Los Blancos boss Julen Lopetegui's close relationship with Thiago could see the Spaniard snub a return to Camp Nou to make the move to the Spanish capital.
Thiago joined Bayern from Barcelona in the summer of 2013, having played over 100 times for the Catalan giants.