His contract with Liverpool expires on June 30th, and Madrid's first match at the Club World Cup is on June 18th, 12 days earlier.
This means Madrid would need to negotiate with Liverpool to release Alexander-Arnold from his contract early.
Real Madrid are reportedly considering offering Liverpool around €1m (approximately £850,000) to release Alexander-Arnold early, as well as paying any outstanding wages owed to him.
FIFA has introduced a short transfer window from June 27 to July 3rd to allow teams to make changes to their squads due to the packed schedule leading up to the tournament.
This window should give Alexander-Arnold time to join Real Madrid, and Liverpool will receive compensation for his departure.