The 23-year-old, who was signed from non league Belper Town in the summer, will spend the next week with the Premier League club.
"I have some contacts at Everton and we felt that the opportunity for Aaron to spend a week at a big club like that was too good an opportunity to miss," Chesterfield boss Paul Cook told reporters.
"It will be a great for Aaron to spend time at a top-flight club, working with some respected coaches and top players. I am sure he will be benefit greatly from the experience."
Chapman is highly regarded in the Football League.