The 25-year-old has impressed for the Reds since joining from Southampton for £34m in the summer of 2016.
Mane's tally stands at 16 goals in 36 appearances for Jurgen Klopp's side, including four in three Premier League games this season.
According to Spanish outlet Diario Madridista, Los Blancos have been monitoring Mane's progress since his move from St Mary's Stadium.
The report suggests that the Real board are impressed with the Senegalese's qualities and could earmark him for a move in the coming months, but Barcelona are also believed to be in the running.
Mane is currently serving a three-match ban after seeing red for his challenge on Manchester City keeper Ederson in Liverpool's 5-0 defeat at the Etihad Stadium.