Vasilyev said the deal for the Colombian, who has scored 28 times in the Primera Division this season, is "on track" but "not yet finalised".
In the interview with RMC radio, the Azerbaijani also dismissed talk of a move for Manchester City's Carlos Tevez.
On Falcao, he said: "It is on track, but it is not yet finalised, there are still details to work out.
"We really hope to complete the signing of this great player. He is one of the best players in the world, we will be very happy when he signs."
Asked what motivation the 27-year-old, one of the hottest properties in world football, had in joining a side just promoted to Ligue 1, Vasilyev said: "He believes strongly in the future of AS Monaco.
"This is a team that has a great history - seven times champion of France and with a project to make a great team. He is very motivated."
There have been reports of a clause in the deal which would allow Real Madrid to sign Falcao from Monaco, to avoid him moving to the Bernabeu directly from their city rivals, but Vasilyev rubbished the idea.
"No, if you sign Falcao for one year, that does not do anything," he added. "There is no clause with Real Madrid, I can say that. There is no other choice for building a team."
On Tevez, Vasilyev added: "Many newspapers talk about it, but this issue has never been discussed."
Monaco, who won Ligue 2 last summer, have already embarked on a spending spree this summer, backed by their Russian owner Dmitry Rybolovlev.
The principality club have signed Porto pair Joao Moutinho and James Rodriguez for a combined fee of 70 million euros (£60million) and also Portuguese defender Ricardo Carvalho from Real Madrid.