Rodriguez is currently the toast of Bavaria with his impressive displays helping Bayern Munich soar towards another inevitable Bundesliga title.
The Colombia international moved to the Allianz Arena from Real Madrid in the summer after snubbing moves to the likes of Manchester United and Chelsea.
But Spanish website Don Balon say Liverpool are determined to succeed where their two Premier League rivals failed.
It is said the Merseyside giants have made contact with Real Madrid to express an interest in signing Rodriguez.
Bayern signed the 26-year-old on a two-year loan deal in the summer.
Yet Liverpool supposedly want to muscle in on their bid to sign the Colombian on a permanent basis.