Mundo Deportivo says the Reds have made a sensational offer to bring Asensio to Anfield.
Liverpool like Asensio "very much" and are willing to go all out to secure his services.
The also say that while Los Blancos will "reject the offer in principle", they be forced to "study" it.
His agent Horacio Gaggioli says Real have already rejected two offers of £130m last month.
"Madrid have received two offers for Marco Asensio, up to €150m (£131m), but the club rejected them," he told Cadena SER. "He has an extraordinary future.
"You can imagine that they (the offers) come from England because they can afford it without any problems.
"They could even raise it."