The left-back is seen as one of the best options for the Reds as they look for cover for Jose Enrique, who has had injury problems in the past, and the fact the 22-year-old can also operate in midfield is an added bonus.
Rodgers wants to take Melgarejo on a season-long loan deal with a view to a permanent move should it prove to be successful.
Any deal is by no means cut and dried though, as the defender is under contract at Benfica until 2018 so they are in no rush and the Reds would also need a work permit for the player which would delay a switch.
Paris Saint-Germain are reportedly ready to launch a summer raid for Chelsea left-back Marcos Alonso.
Terriers midfielder Phillip Billing has spoken about the whirlwind nature of his move from Denmark to Huddersfield as a teenager.