The Real Madrid midfielder has consistently been linked with move away from the Bernabeu after falling out of favour under Zinedine Zidane.
Real are thought to be willing to exchange him in return for Chelsea’s Eden Hazard, despite the Blues insisting he is not for sale.
Manchester United are also thought to be ready to make an offer in the transfer window with Jose Mourinho keen to bolster his midfield.
James Rodriguez is attracting interest from Liverpool, Chelsea and Man Utd
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But The Mirror claim Liverpool hope to complete a deal for the £65m-rated star with the club already making contact with his representatives.
However, their attempts to prise him away from the Spanish capital rest on whether they can secure Champions League football.
Rodriguez, who is now thought to be open to the idea of a move, will likely favour a switch to a club that is competing in Europe’s elite competition next season.
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Jurgen Klopp's Liverpool have reportedly already contacted James Rodriguez's representatives
A source close to Real told The Mirror: “It seems Zidane is not satisfied with the player this season and has already asked for reinforcements in this position for next season.
"The player isn’t happy and it now appears a move in the summer is very feasible.”
Liverpool currently sit fourth in the table, four points above fifth-placed Manchester United.