The 25-year-old, who has been on the Potters' radar since last summer, passed a medical last week and has now agreed personal terms.
Speaking to the club's official website, City chief executive Tony Scholes said: "Moritz is a player we have been trailing for some time and was someone Mark Hughes wanted to bring into the club.
"Moritz came over to England last week to undergo his medical and discuss terms and we have now been able to conclude his transfer. He's a very highly-regarded player who broke into the Austrian senior side last year and he adds to the defensive resources at our disposal."
Swiss-born Bauer began his career with Grasshoppers Zurich before joining Kazan in 2016, playing 37 league games in total for the Russian side.