Diaz, 28, had expressed his desire to join Manchester City, led by manager Pep Guardiola, after becoming aware of their interest in him.
Liverpool was hesitant to agree to the transfer, but inquired about Alvarez's availability as part of the deal.
However, Manchester City rejected the proposal, as they did not want Alvarez to join a Premier League rival.
Alvarez eventually signed with Atletico Madrid for £82 million and went on to have a successful season, scoring 29 goals and providing 7 assists in 54 games.