"Things are progressing well, but it's not finalised yet, there are still some details to sort out," Vadim Vasilyev told French radio RMC.
"We really want to complete the transfer of this great player. He is among the best players in the world, we will be very happy when he signs. He really believes in the future of Monaco."
Falcao, with 28 league goals this season, appeared to be saying goodbye to Atletico fans on Sunday after their final home La Liga game.
Reports have suggested Falcao, 27, who has a price tag of around 60 million euros, is poised to join Monaco, who have just won promotion back to France's top flight and have an ambitious billionaire Russian owner.
Monaco have already signed Joao Moutinho and James Rodriguez from Falcao's former club Porto for a combined fee of 70 million euros, as well as Real Madrid centre back Ricardo Carvalho as a free agent.
Monaco are at odds with French football authorities over tax exemptions but they have attracted many leading players helped by the fact that, under the principality's laws, foreigners do not pay tax on their wages.
The French League (LFP) has ruled that the club must move their head office to France by the beginning of June 2014, meaning players would pay tax in France.
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