New Los Blancos manager Julen Lopetegui is a big fan of the Senegal international and wants him at the Bernabeu.
Mane impressed last season for Liverpool, helping Jurgen Klopp’s side to the Champions League final.
Spanish outlet Diario Gol, known for their bold transfer claims, say Real have made a huge proposal to the Reds.
Los Blancos are prepared to pay £70m and include goalkeeper Keylor Navas in a deal to sign Mane.
That would make the total value of £87m (€100m) that Liverpool are said to be demanding to let the 26-year-old leave Anfield.
The Reds could be tempted as they are looking for a new goalkeeper this summer.
That’s following two horrendous errors from Loris Karius in the Champions League final defeat to Real Madrid.
And Navas would certainly have the credentials to take place in Klopp’s side after playing at the Bernabeu since 2014.
Real Madrid president Florentino Perez will not make any move for Mane though until both stars have been eliminated from the World Cup.
Mane netted 20 times and provided nine assists in all competitions for Liverpool last season.
But he was overshadowed by team-mate Mohamed Salah who scored an incredible 44 goals across all competitions.
Real Madrid are looking to strengthen their squad after being well beaten in La Liga by arch-rivals Barcelona.
Los Blancos did create history though by winning the Champions League for the third straight season.