The Montenegrin defender has been chased by the club for some time.
Savic told City’s website: “I watch the Premier League all the time and I know it’s the place to be so I am just looking forward to helping this club achieve great things. We have an incredible stadium and I’ve seen how passionate the City fans are many times on TV.
“Montenegro play England later this year so I’m expecting to have some great banter with the England lads. I’m just really happy to be a City player.”
