The winger, who hit two late season hat-tricks for the rejuvenated Reds, wants to return to his homeland and has been in touch with the Argentine club's chairman about a move
The 30-year-old winger, who joined the Reds on a free transfer from Atletico Madrid in the 2010 winter window, represented the Rosario-based side for three seasons after emerging from their youth team before moving to Spain.
The Argentine, who struggled to cement a spot in the starting XI before finishing the season in goalscoring form under Kenny Dalglish, has stated that he has always wanted to return to his homeland and has been in touch with his former club about a move.
He told La Capital: "When I left here, I knew I was coming back.
"The president [William Lorenzo] called me and I said yes. Now it depends on the contract I have in Liverpool, maybe that can be loosened a little."
Dalglish is believed to be interested in adding to his options down the flanks, with Aston Villa's Stewart Downing repeatedly linked with a move to Anfield, and Rodriguez could well fall further down the pecking order.