The Sunday Mail claims that the Spanish giants are tracking the Scotland Under-21 star and have watched him twice already this season.
Real chief scout Julen Lopetegui confirmed: "He is now one of 10 young players we are watching across Europe. It's a complicated process and we know it will be a hard choice."
Madrid sporting director Predrag Mijatovic has received a glowing report on the striker who has scored 34 times in 133 appearances for the Edinburgh side. A Bernebeu source added: "It's true Fletcher is in the second phase now. We will watch him again before making a definite decision."
The starlet made a strong impression on Spanish scouts during Scotland's run in the Under-19 European Championships two years ago. Scotland lost out 2-1 to Spain in the final but under-21 gaffer Inaki Saez admitted he is a fan of the forward.
"During the last couple of years the level of young players in Scotland has increased. Fletcher is proof of that. He has a big future and I can't understand why he isn't playing for Celtic or Rangers."