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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Since Guus Hiddink arrived at the helm, Chelsea have improved considerably after a tricky start to the campaign.
Newcastle United are staring down the barrel of relegation to the Championship for the second time in the past eight years.