Colombia international Falcao scored over 100 goals in his first three seasons in European football and was widely touted as one of the best strikers in the world, but suffered an anterior cruciate ligament injury at Monaco in January 2014.
He recovered after undergoing knee surgery and secured a high-profile loan move to United but failed to light up the Premier League, scoring just four times in 26 league appearances in the 2014/15 season.
The 30-year-old spent the following season at Chelsea where he was even less successful, scoring just one goal in 10 games.
Falcao has since returned to Monaco to rebuild his career and Vasilyev believes both United and Chelsea were too impatient to get the best out of the Colombian.
"Both clubs in England failed to relaunch him," he told the Daily Mail.
"I don't know whether Manchester United or Chelsea did not have the patience or the desire to do that but I am quite sure this year we will see Falcao at his previous best.
"He is the same Falcao as ever. We've seen that in pre-season. He scores goals, he's there and he is motivated, we are motivated and I think this will be his year."
Monaco are top of Ligue 1 with three wins in their four league fixtures to date, and Vasilyev says the French club's coach Leonardo Jardim will get the best out of Falcao this season.
"If you sign a player of this magnitude after a big injury it takes time and games to get him back into shape," he added.
"This is what we will do and what Manchester United and Chelsea didn't do, failed to do or didn't want to do.
"We all see the qualities are there and it is a matter of time. You will see a very different Falcao this season to the one you saw in England."