Mbappe joined PSG from Monaco on a season-long loan after breaking on to the scene with a stunning run of form in their run to the Champions League semi-finals in the 2016-17 campaign.
The France forward is set to complete a permanent move to PSG for €180 million (£158m/$210m) ahead of the new term but Mbappe could have moved elsewhere.
Former PSG coach Unai Emery suggested earlier this month that Mbappe wanted to go to either Madrid or La Liga rivals Barcelona, rather than staying in France.
But Vasilyev says that was not the case, with Madrid having an offer accepted for the teenager only to be rejected by the player, in favour of joining PSG.
"I remember we were training when we got the call from Real Madrid and accepted the €180m offer for Mbappe," Vasilyev told Canal+.
"Later, I saw him and told him 'we have received an offer that was impossible to refuse from the club of your dreams so tell me your answer'. We never wanted to make the choice for him, it was up to him to make his decision.
"He told me that he wanted to go to Paris and, after that, there were so many details that had to be resolved that, until the final moments, I doubted the deal."
Mbappe has now emerged as a transfer target for Manchester City, as revealed by Goal, despite still having the option to wrap up his permanent move to PSG.