Salah only joined the Reds during the summer, but his performances so far this season have piqued the interest of a number of Europe's top clubs, with Madrid understood to lead the way.
The Egypt winger is the Premier League's top scorer with 13 goals from just 16 games, while in all competitions he has 19 to his name already.
Spanish media outlet Don Balon claims that Madrid are serious about their pursuit of Salah and will try to coax him away from Anfield in January as they look to salvage their season.
Los Blancos currently sit fourth in La Liga, eight points adrift of leaders Barcelona, while they also face a serious threat to their Champions League crown having been drawn against Paris Saint-Germain in the last 16.
Madrid are reportedly ready to offer Lucas Vazquez as part of any deal to bring Salah to the Bernabeu, although Liverpool are unwilling to consider any bids for the winger.
Vazquez has started just seven games for Madrid across all competitions so far this season.