The Italy international was heavily linked with a move to the San Siro side, but a transfer seems to be out of the question for now.
D'Amico revealed earlier this month that Milan had made an inquiry about the availability of the 27-year-old, with Criscito himself stressing that he would happily take a pay cut in order to return to Serie A.
WORLD CUP: Beautiful people | Germany homecoming
However, the defender's current club has no intention of letting go of Criscito at this stage.
"Criscito will definitely stay at Zenit," D'Amico told Radio Kiss Kiss. "Villas-Boas knows him very well, both from a personal point of view and as a player. He does not have any intention to sell Domenico and will even make him one of the team's key players.
"Zenit don't need any money. They only sell players whom they deem surplus to requirements."
The left back started his professional career with Genoa and also enjoyed a spell with Juventus before leaving Serie A for Zenit St Petersburg in June 2011.
Criscito has a contract with Zenit until the summer of 2016.