The Spain international's contract at Stamford Bridge is due to expire this summer, and Barcelona have been strongly linked with a move for the versatile defender.
A recent report claimed that the Catalan giants were confident of securing a deal for the 32-year-old, who has made 27 appearances for Thomas Tuchel's side this season, contributing three assists in the process.
However, according to Defensa Central, Real Madrid are also interested in the Spaniard and are preparing to battle for their Clasico rivals for his signature.
Azpilicueta has previous experience of La Liga, having turned out for Osasuna's first team between 2007 and 2010 before making the move to Marseille.
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