Tombides was diagnosed while away on international duty with Australia at the Fifa Under-17 World Cup in Mexico.
"Dylan is receiving the best possible care," club doctor Sean Howlett told West Ham's official website. "All the doctors involved have advised Dylan that he will make a full recovery in terms of his illness and his ability to return to his football career.
"He is undergoing the standard series of treatments for his condition which is a course of chemotherapy."
Hammers academy director Tony Carr added: "Everyone at the club is right behind Dylan and we all wish him a speedy and healthy recovery.
"The doctors are all very positive and he is in the best possible hands. We look forward to seeing him back with us when he is ready."