The Russia international forward is set to join the cash-rich club, who are continuing their summer spending spree, which has included Igor Denisov for €15m
The forward - who can play in attack or out wide on the right - has expressed his desire to join the cash-rich Russian club, according to a member of Dinamo's board of directors, and that money and ambition are the reasons for his departure.
"The decision to leave Dinamo was made by Kokorin, not us," Vasily Titov told Russian news agency ITAR-TASS.
"We did have an offer from Anzhi but not for €20m but for a €19m. This amount was written into the contract as the clause for the player. And, in accordance with the contract, only the player himself can decide whether to move away from Dinamo to another club or not.
"Kokorin said he wanted to go to Anzhi. This is his personal wish. He is motivated by two things. The first is the growth of his sporting ambitions which he believes in Anzhi can fulfil. The second is a more favorable salary."
The 22-year-old Russia international will become the fourth official signing of the summer transfer period for Anzhi coach Guss Hiddink following the arrivals of Igor Denisov, Mikhail Kerzakov and Alexei Ionov.
Christopher Samba is also due to imminently return from relegated English club QPR, fewer than six months after he left the Dagestan club and for an identical fee to what the Championship club paid in January.
Kokorin, who scored 10 goals in 22 leagues games in the 2012-13 campaign, joined Dinamo in 2008 from Lokomotiv Moscow and has a contract with the capital-based club until 2016.