Ronaldo secured a move to the Italian side earlier this month and leaves behind some big boots to fill, having scored 450 goals in 438 appearances for Los Blancos, winning two La Liga titles and four Champions Leagues.
Asensio is seen as one of the brightest prospects in world football and it appears he will succeed Ronaldo and take on Madrid's iconic number seven shirt, according to captain Sergio Ramos.
He said, referring to Asensio and quoted by Diario Gol: "He will take Cristiano Ronaldo's seven."
It is yet to be seen whether Madrid will make a Galactico-esque signing this summer, with the club favouring signing younger players in recent years, but Chelsea duo Thibaut Courtois and Eden Hazard have been heavily linked following their World Cup excursions.
However, the report also claims Asensio could be used as a potential makeweight for a deal for Hazard, as Real Madrid president Florentino Perez will favour a move for the Belgium international instead of keeping his Spanish counterpart at the Bernabeu.
Former Spain manager Julen Lopetegui has recently taken the reins at Real Madrid and is reportedly a big admirer of Asensio, who has 15 caps for the national team, and it seems unlikely that the former Mallorca and Espanyol attacker will be allowed to leave this summer.
Real Madrid have already had a busy transfer window, bringing in Alvaro Odriozola, Andriy Lunin, and Vinicius Junior permanently.