Mombaerts, 64, has previously been in charge of the likes of PSG, Toulouse and Le Havre, while his most recent job was at the helm of Yokohama F. Marinos from 2015 to 2018.
The former France Under-21s and Under-18s coach takes over a City side that lost in the Elimination Finals in 2018-2019 before the A-League club opted not to renew Warren Joyce's contract.
"I am delighted to join Melbourne City at this exciting time for the club. I am here to achieve two main objectives – to implement a style of play that gives the team a clear identity, and in doing so to win silverware," Mombaerts said in a statement on Thursday.
"Having worked in both Japan and in Manchester with City Football Group, I know the City style of play inside out. It is my challenge now to bring my experience to Melbourne, which has a different set of players and a different football culture, and to apply it so that we can win.
"It is also very important to me that fans can feel proud of how we play, and that they will enjoy the whole experience watching the game.
"We have some fantastic young talent, some exciting new signings and a solid existing squad. My focus now is on our pre-season preparation, together with my coaching team, who provide a fantastic combination of local and international experience. Our hard work begins today."
While several players have departed since the end of last season, City have also signed former Bolton Wanderers winger Craig Noone and Australian full-back Scott Galloway.