The 20-year-old right-back scored his first goals for the club when he netted twice in Saturday's 5-1 win over Hannover, adding to the seven assists he has already made this season.
But former Dortmund star Sebastian Kehl, who is now the head of licencing players at Dortmund, says it is unlikely Madrid will be willing to let him go.
"Unfortunately, we only loaned him for two years and it's going to be difficult to sign him," Kehl told Sport1.
"Real Madrid are also following his development."
While Hakimi's stock is rising at Dortmund, the club are in danger of losing an established fan favourite in Shinji Kagawa.
Dortmund sporting director Michael Zorc has said the club are willing to let him go, but Kehl says the situation is heartbreaking for him.
"If I see Kagawa, it hurts my heart," Kehl said when asked about his former team-mate.
"He came to us, he wants to go, he wants to stay in Europe and he prefers a move to Spain, but in the winter break it's difficult to find a suitable club."