Madrid were interested in Mbappe prior to the French sensation swapping Monaco for Paris, and the LaLiga powerhouse continue to be linked with the 20-year-old forward.
"Why did he not sign at Real?," Vasilyev told Telefoot. "He told me: 'Vadim, I think it's too early, I'm Parisian, I do not want to leave my country like that, I want to become a great player here. Real Madrid will wait, that's what I think'.
"I told him, 'Kylian, you're right'. He wants to win. And of course the Champions League must be won for a player like that. If he wins it with Paris, it's better."
Vasilyev added: "They [Real Madrid] will knock on your door. The entire Estadio Santiago Bernabeu will applaud you when you get there."
"He remains the same person, the same Kylian. It's just that he has gained importance, power in the global landscape," Vasilyev said.
"I know Cristiano Ronaldo quite well, and he always wants to break all records, go further. Kylian is the same."
Mbappe, who won the Ligue 1 title in his final season at Monaco, has flourished with PSG in the French capital.
He has claimed back-to-back Ligue 1 trophies, as well as Coupe de France, Coupe de la Ligue and Trophee des Champions silverware.
After scoring 39 goals across all competitions in 2018-19, Mbappe has netted nine times for Thomas Tuchel's PSG this season.