The Red Devils youngster joined from Atletico Madrid in 2020, and has played for Argentina at youth level.
According to Spanish outlet Relevo, Real are already monitoring the talented forward, who has burst into the United first team this season.
Real are interested in building for their future, which means that Garnacho and Brazilian forward Endrick are on their hot list.
However, United would loathe to sell, and recent reports from Fabrizio Romano did suggest the club was aiming to renew Garnacho's contract this season.
The forward does regularly visit his family in Madrid, which means the lure of home may eventually be too great for him to resist.