The Portuguese superstar only returned to Old Trafford in the summer, the 36-year-old already scoring 12 goals during his second spell at the club.
However, the appointment of Ralf Rangnick as interim manager and the German's subsequent playing style has led to suggestions that Ronaldo may not retain a place in the starting lineup.
According to Football Insider, Los Blancos are monitoring the situation ahead of potentially making a bid to secure their own reunion with the player.
The outlet cite a recruitment source as claiming that Real may look to take advantage of the situation if Ronaldo does not remain as first choice under Rangnick.
Ronaldo spent nine years of his career at the Santiago Bernabeu between 2009 and 2018, scoring 450 goals in all competitions and winning 15 trophies.