Cruyff indicated last month he would take on the assignment, his first official position with Ajax since being sacked as coach in 1988.
"There is a professional difference of opinion and not a personal conflict because I want to bring back the level of the youth department to where it belongs," the 60-year-old former European Player of the Year said in a statement issued by his foundation.
The differences between Cruyff and the new technical staff led by Van Basten, who will take over from Adrie Koster as coach of Ajax after leading Netherlands at Euro 2008, were too great, added the foundation.
"There are drastic measures necessary. I made a clear plan but my vision is not shared by those who should execute it," said Cruyff.
"What Johan wanted to do for Ajax's youth department was going too fast for me. I am not so dissatisfied with how things are going now," said Van Basten.
Ajax said Cruyff had given back the assignment and that Cor van Eijden, chairman of the club's member council, was 'surprised' at the news.
The team are second in the league, six points behind PSV Eindhoven.
Ajax also failed to reach this season's UEFA Cup group stage, their worst European campaign in 20 years.
Cruyff started his career at Ajax in 1964 before joining Barcelona in 1973.
He returned to the Dutch club in 1981 for a two-year spell and in 1985 he started his coaching career, taking Ajax to victory in the 1987 European Cup Winners' Cup.
Van Basten began his playing career at the Amsterdam team in 1981 and remained there until he joined AC Milan in 1987.