Van Basten stepped down from his role earlier this week to become an assistant due to health problems and Pastoor was identified as his likely replacement.
The 47-year-old was appointed assistant to Van Basten in the close season, but rather than stepping up to replace his former boss Pastoor has left the club with immediate effect.
The club revealed in a statement that the former assistant had been their number one choice to take over as head coach, but they will now look elsewhere for Van Basten's replacement.
"In the past few days we have been in conversation with Alex," said director of football affairs Earnest Stewart. "After the decision to make Marco van Basten assistant coach we had the intention to promote Alex into the position of head coach.
"We made him an appropriate, improved offer. However, he could not agree to [it].
"[During] the process it became clear that Alex was not going to act as an assistant under a new head coach.
"After the last call it was concluded that it was better to go apart. We wish Alex every success in his future coaching career."
AZ have started the new season with two wins from their five matches and are yet to win at home in the Eredivisie.