The 25-year-old has been in impressive form for Jose Mourinho's side at the start of the 2021-22 campaign, scoring six times and contributing two assists in eight appearances.
Pellegrini is believed to be in advanced talks with Roma over a new deal, but there has still not been an announcement, and the Italian's current contract is due to expire next June.
According to Fichajes, Los Blancos are monitoring the situation and will approach the midfielder in January if he still has not committed his future to his Serie A outfit.
Pellegrini has scored 27 goals and registered 36 assists in 160 appearances for Roma in all competitions.
Real Madrid are believed to be keen to sign a new central midfielder in 2022, with Luka Modric coming to the end of his time in the Spanish capital. body check tags ::