The Mirror reports that Rodgers is keen to bring in Melgarejo as back up for injury prone Jose Enrique.
The 22-year-old was a regular starter with Benfica last season, but is keen on a move to a bigger league this summer.
Rodgers is keen to sign Melgarejo on an initial year long loan deal, with a view to a permanent move to Anfield.
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Liverpool have unveiled their new home kit for next season, sticking with the hugely popular retro styling of the current shirt.
The reputation of the Premier League means it is the most popular and one of the most watched leagues in the world.