Van den Brom, who last worked at Belgian champions Anderlecht but was fired in March, had a verbal agreement with the club which would be firmed up with a proper contract in the coming week, a statement from AZ said.
He will not be on the bench for Saturday's visit to ADO Den Haag, where Dennis Haar and Van Basten will lead the side in a caretaking capacity, but will start work next week.
Van den Brom is a former player at Ajax Amsterdam and Turkish club Istanbulspor, winning two caps for the Netherlands, and has coached at Den Haag, Vitesse Arnhem and Anderlecht, with whom he won the Belgian title in 2013.
Former European Footballer of the Year Van Basten quit earlier this month after suffering heart problems brought on by stress. He negotiated to stay at the club as assistant, swapping duties with Alex Pastoor.
However, Pastoor was removed within 48 hours of his promotion after the club accused him of leaking details of Van Basten's medical condition to Dutch media.
(Reporting by Mark Gleeson; Editing by John O'Brien)
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