The striker has recently switched agents to Pini Zahavi as he is reportedly eyeing a move away from Bayern Munich.
Chelsea and Manchester United are chasing the Poland international, who is the top scorer in the Bundesliga with 23 goals.
But, according to AS, Lewandowski wants to join Real Madrid this summer over anyone else.
The 29-year-old is said to have his heart set on a move to the Spanish capital and he has asked Bayern to let him go.
However, the German champions are desperate to keep their star man and are trying to persuade him to stay.