It is thought that the Red Devils were ready to offer the 17-year-old centre-back, who has been compared to compatriot Daniel Agger, a contract at Old Trafford after undergoing a successful trial period.
However, it seems that the Brondby youngster, who has previously captained his national U17 side, has chosen to move to the Amsterdam ArenA.
"It will be exciting and I am happy to look at what Ajax can offer me," Boilesen told Danish publication Sporten.
"There is obviously a positive impact on the decision when there is already a Dane there," he added, referring to Ajax's Danish international midfielder Christian Eriksen.
Ricky Brooks, Goal.com