The Brazilian has a contract with the Blaugrana until 2015 but is now said to have put the Russian club on alert by admitting that he would consider moving on in the summer
The Brazil international has established himself as one of the key components in Pep Guardiola's side since arriving from Sevilla in 2008 and last year signed a new contract with the Catalans, which ties him to Camp Nou until 2015.
However, big-spending Anzhi, who are funded by billionaire Suleyman Kerimov, are preparing a massive bid to tempt Dani Alves to Russia, according to Radio Marca.
The 28-year-old, who has been linked with Premier League duo Chelsea and Manchester City in the past, has already won three Liga titles and two Champions League crowns since joining Barca and his agent had claimed before the player signed his contract extension last March that the club had been undervaluing his client.
Even though Dani Alves was left out of the starting line-up for Tuesday's Champions League semi-final second leg with Chelsea for tactical reasons, it is doubtful whether the Catalans would seriously consider any offers for the defender.