James is currently on loan with Bundesliga leaders Bayern Munich, having signed a two-year temporary deal to join the Bavarian giants last summer following an unsuccessful spell in Spain.
However, Don Balon reports that Liverpool are looking at the possibility of cutting his stay at the Allianz Arena short and signing him on a permanent basis at the end of this season.
Rodriguez's loan arrangement with Bayern complicates the move, but Liverpool will reportedly open negotiations soon in order to allow enough time to reach an agreement with both Madrid and Bayern before the summer.
Any deal would rely on Bayern willingly terminating the Colombian's loan early, while Madrid are understood to want to recoup as much of the £70m they paid for him as possible.
Liverpool are expected to face competition from a number of other top European clubs for the signature of Rodriguez, who has scored three goals and laid on four more in 13 Bundesliga appearances this season.