The Reds boss is thought to be a huge admirer of the France international, whom the club attempted to lure to Anfield over the summer.
However, their approaches at the time were rebuffed by Real boss Carlo Ancelotti, who tied the marksman down to a new deal at the Bernabeu.
But, this has not deterred Rodgers, whose attacking issues are well known.
As it stands, Rickie Lambert, with two goals, is the only orthodox Liverpool striker with more than one effort in the Premier League to his name this term, with the likes of Mario Balotelli and Fabio Borini having struggled in Daniel Sturridge’s absence.
The ex-Lyon man – who scored at Anfield in the Champions League earlier this season – is seen as the answer, with his scoring record having been impressive throughout his career.