According to a report in Spanish newspaper AS, Axel Werner, the current second-choice goalkeeper at the Wanda Metropolitano, is expected to spend next season on loan in order to get more game time - so Diego Simeone will need a new understudy for his current first choice Jan Oblak.
Adrián is one of three possible targets mentioned in the article - another one being Manchester City's Claudio Bravo, who has been eclipsed by Ederson at the Etihad. Getafe veteran Vicente Guaita is another potential target for Simeone.
22-year-old Werner is generally believed to have an exceptionally bright future, but he made just three appearances for Los Rojiblancos last season, two of which were in his club's triumphant Europa League campaign. Whoever replaces Werner will struggle to oust Oblak as first choice keeper, as the Slovenian is widely regarded as one of the most dependable goalkeepers in Europe.
This may prove to be a deterrent for Adrián, who grew increasingly frustrated with his lack of matches for West Ham. He made just 19 Premier League appearances for the Hammers last season, as former manager David Moyes favoured Joe Hart - in spite of several high-profile blunders by the England man.
In fact, the Spanish keeper became so furious with Moyes that he reportedly gave his club an ultimatum - either Moyes had to leave, or he would.
Adrián would likely have considerably less game time under Simeone than under Moyes - although the lure of UEFA Champions League football may also appeal to the 31-year-old Hammers keeper, who is yet to feature in Europe's elite competition.