Premier League rivals Arsenal are the other side said to be monitoring the Argentine, with both clubs eyeing up a summer move.
The Reds could view Rodriguez as a replacement for Gini Wijnaldum, who could leave the club as a free agent this summer and have ‘followed up’ on the 26-year-old with Betis.
In those appearances, Rodriguez has earned an average rating of 6.91 and completing 84 per cent of passes accurately (Sofascore).
The report claims that the midfielder has a £69m release clause written into his contract, although his market value stands at just £16.2m (Transfermarkt).
Jurgen Klopp is relatively well-stocked in the midfield department, even if Wijnaldum does leave at the end of the season.
The signing of Thiago last summer and having Fabinho back in his recognised midfield berth should be enough options for next term.
That could change should another leave the club, although even then forking out a £69m release clause for a position not considered a priority, looks highly unlikely.