The USA international is available from Dortmund after new contract talks fell through last week. With his deal running to 2020, Pulisic could be sold this month.
Don Balon says Real are in contact with Pulisic via intermediaries and have been assured he would welcome a move to Madrid.
Indeed, it's been made clear the midfielder would prioritise a move to Real Madrid ahead of anyother offer.
Ironically, Chelsea are expected to bid for Pulisic this week, with Blues owner Roman Abramovich informing BVB of his plans last weekend.