The Colombia attacker has ambitions of eventually playing in LaLiga, where Atletico Madrid are also keen.
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Mundo Deportivo says Diaz currently favours a move to Barca and is attracted to the prospect of playing for a Spanish-speaking club.
Barca are willing to try for Diaz next summer, knowing Liverpool rate him at around €75m.
Diaz has five goals and two assists in the Premier League this season.