Reports in Mundo Deportivo had suggested that the Argentina international had rejected earning €30m [£24.5m] a season with the Russian club, after his buy-out clause was allegedly met.
However, their marketing consultant German Tkachenko quashed the story as totally false, and insisted that the side do not operate that way in the transfer market.
"There isn't a shred of truth in it," he told Russian newspaper Sport-Express.
"Asking a question about this, let alone writing about it, shows a lack of understanding about how Anzhi's sporting model is designed and what its principles and priorities are."
Spanish media reported that Messi not only dismissed the Russian offer, but duly signed a new contract tying him to the club until 2018.
Former Camp Nou striker Samuel Eto’o, who played with Messi at Barca, made the switch from Spain to Russia, and is currently the highest paid footballer on the planet, on a reported €20m [£16.3m] per year.