Savic can play as a right back or centre back and joins former Arsenal defender Gael Clichy at Eastlands, The 20-year-old was granted a work permit by the Home Office thanks to his prominence in the Montenegro national team.
"I can't wait to play alongside them and all the other fantastic players we have here.
"It is a dream come true and I'm especially looking forward to meeting Edin Dzeko and Aleksandar Kolarov because they both speak my language and come from the Balkans, too.
"Aleks has already phoned me to welcome to City which makes me feel at home already.
"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."
