Zinedine Zidane was keen to sign the Senegal star, 26, last summer but the move fell apart when he opted to quit Los Blancos.
Now, Marca is reporting the Frenchman will launch a £73million bid this summer.
Real have also been heavily linked with Chelsea’s Eden Hazard.
But the Blues’ two-window transfer ban could see them hold on to the Belgian, forcing the Spaniards to turn their attention to Mane.
The speedster has been outstanding for the Kop this term – bagging 20 goals in all competitions, level with Mo Salah.
Liverpool are financially robust and, with Mane’s current deal running for another four years, it would take a huge fee for Kop top brass to even consider doing business.
Jurgen Klopp’s side are two points clear of Manchester City after their win at Fulham.
The Reds return to Premier League action at home to Tottenham before travelling to Southampton.
Mane scored a brace against Bayern Munich to set up a Champions League quarter-final against Porto.
Klopp is looking to go one better than last year when Real beat his Reds in the final.