The Wanderers winger will move to Australia when his contract expires at the end of this month.
Noone played 65 games for Bolton after signing on a free transfer from Cardiff City, scoring twice.
He joins former Bolton youth product Scott Jamieson at AAMI Park, and will be facing the likes of Adam Le Fondre and Jem Karacan this season.
"I'm excited to join Melbourne City and test myself in the A-League,” Noone said. “I want to win and be competing for silverware.
“This is an ambitious club that wants to win trophies while playing an exciting style of football.
"I know a few players in Australia and know the quality is good. I'm expecting it to be physical and fast-paced and can't wait to step out in front of the fans for the first time at AAMI Park.”
Melbourne are a sister club to Manchester City and finished fifth in the league last season.
They are currently without a coach following the departure of another ex-Wanderer, Warren Joyce.