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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Summer 2016 is set to be a very exciting time for football fans, as this is when the long awaited Euro 2016 tournament kicks off.
Since Guus Hiddink arrived at the helm, Chelsea have improved considerably after a tricky start to the campaign.