Just 20, Havertz has emerged as one of the hottest transfer properties in Europe and is being linked with a big move away this summer.
BILD says Real have already acted and offered €80m for the youngster - which Bayer immediately rejected.
Bayern Munich are also interested in the midfielder - which was confirmed by honourary president Uli Hoeness last week.
Just this week, Bayer assistant coach Marcel Daum declared: "Everyone is asking about Kai Havertz.
"If you see him play, it tells you everything.
"For me, he has the elegance of Zidane. If you see his movement, he's very elegant, he's always calm, he has great passing ability.
"What a lot of people don't see, because we watch him every day, is that he's pretty fast. That's a big bonus.
"He's a great player; he's scoring goals, he's great in the air and he's pretty fast. It's up to him how far he can go.
"I can't say he will be the best German player ever, but he has the talent and there are no limits. He's displaying signs of Zidane."