Haaland joined Dortmund from Red Bull Salzburg in January, but has made an immediate impact in Germany by scoring 12 goals in his first 11 appearances.
The 19-year-old chose BVB because he thought it would be the ideal place to continue his development, but has quickly shown he has enough talent to play for any side in Europe and his performances have alerted Real Madrid.
Los Blancos signed Luka Kovic for €60 million last summer as back-up to Karim Benzema and the 22-year-old has been described as "the future" by coach Zinedine Zidane, but the Serbia striker has scored just twice in 24 appearances so far.
Meanwhile, Karim Benzema's form has dipped dramatically since Christmas and Madrid struggled for goals prior to the current enforced break due to coronavirus.
Haaland, who moved to Dortmund for just €20m plus agent fees in January, has a clause allowing him to leave the Signal Iduna Park for around €75m (£50m), although it is unclear if that can be activated this summer.
His agent, Mino Raiola, told Marca in an interview last week that he would look to take one of his players to Real Madrid this summer, although he may have been referring to Manchester United's Paul Pogba.
"He has a lot to improve yet. He'll be at Dortmund for as long as he needs to be," Raiola said of Haaland. "It's true that he is a footballer who is anticipating the stages of growth faster than you might think.
"I don't think that when he left [Red Bull] Salzburg it was the right time, but... I don't think he'll be leaving Dortmund this summer.