Mbappe initially joined Paris Saint-Germain on loan from Monaco in 2017.
"He said 'Vadim, deep inside me, I feel it is too early'," Vasilyev told Telefoot.
"'I have only played for a year in my country, I'm Parisian, I don't want to leave my country like that.
"'I want to become a great player here. Real Madrid will wait, that's that I think'."
Vasilyev is still close to Mbappe and in a recent meeting the the youngster.
"He is still the same person," Vasilyev said. "He has grown, gained influence and notoriety.
"If he doesn't play, he feels uncomfortable and even sick."
The Russian though, was clear in that he is certain Mbappe will eventually strut his stuff for Real Madrid.
"They [Real Madrid] will knock on your door," Vasilyev insists he said to Mbappe.
"The entire Estadio Santiago Bernabeu will applaud you when you get there."