The 54-year-old, who is now the manager of Napoli, won the Champions League, Super Cup and the FA Cup during a six-year spell at Anfield.
Before leaving the Merseyside outfit in 2010 to take over at Inter Milan, Benitez is said to have been contacted by Real president Florentino Perez about taking over at the Bernabeu.
"Rafa was for two consecutive years given the chance to become coach of Real Madrid," Quilon told Calcionapoli24.it. "We received a very interesting proposal from the Blancos five years ago and the year before that.
"In order to convince him to accept the offer, Florentino Perez flew out in person with a private jet to England. He tried, without success, to convince Rafa's wife Montse to follow her husband in this new adventure.
"However, as Rafa had just renewed his contract with Liverpool and is a man of his word, he turned down the offer from his beloved Real Madrid despite the fact it was the best economic and sporting proposal he'd ever received."