That is according to Spanish news outlet Marca, who are suggesting that new Los Blancos boss Julen Lopetegui is considering a move for the former Barcelona ace, but as a potential replacement for Mateo Kovacic.
The Croatian has expressed dissatisfaction over the minutes he was afforded last season and could look to engineer a move elsewhere in the summer if regular playing time cannot be guaranteed.
Real are very much against Kovacic leaving the Bernabeu but, per the report, Thiago has been identified as a splendid alternative and are hoping that Lopetegui's relationship with the player will work in their favour if he is pursued.
The ex-Spain boss worked with the attacker during his time in charge of the international side and he was made to feature heavily, generally performing at an admirable level.
Lopetegui has since been appointed manager at Madrid, a move that cost him the Spain job. But he is already keen on getting to work as he aims to improve his new side's domestic position next season.
Meanwhile, Kovacic's representatives are understood to have told Madrid they should consider all offers for the Croatia international. It isn't expected that Luka Modric and Toni Kroos will be displaced in the team next season, yet the Croatian gave a good account of himself when called upon to play in lieu of either of the aforementioned during the recently closed campaign.
As reports in Spain would have it, Alcantara could possibly be persuaded to return to La Liga if Kovacic departs. But his links with Barcelona could be a huge hindrance.