The defender may be ready to move on from the Anfield club after Jurgen Klopp departed.
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Per BILD, Real will declare their interest in Alexander-Arnold and try to bring him to the Bernabeu.
Long-term right-back Dani Carvajal is getting on in age and may benefit from the competition.
Alexander-Arnold is also said to be keen to make the move to take a different step in his career.
However, Liverpool boss Arne Slot does want to hold onto his creative star this summer.