Lennon, whose father split up with his mother Cynthia when he was five years old, paid 25,000 pounds ($39,030) at a London auction earlier this month for a part of his family heritage that became world famous.
McCartney wrote the song for Julian, originally calling it "Hey Jules" with the poignant lyric "Take a sad song and make it better".
"He has a few photographs of his father but not very much else," Julian Lennon's manager, John Cousins, told The Sunday Times when revealing the mystery buyer at a Beatles memorabilia auction.
"He is collecting for personal reasons. These are family heirlooms if you like," Cousins told the paper.
John Lennon left Cynthia for the Japanese artist Yoko Ono. He was murdered by a crazed gunman outside their New York apartment building in December, 1980.