The 31-year-old winger has been with Wanderers for the last two seasons following a move from Cardiff City.
Noone has made 65 appearance in all competitions and scored two goals – a memorable winner against Derby County last season in which he celebrated in the stands with team-mate Stephen Darby, and another vital strike at Barnsley as Bolton were scrapping for Championship survival in April 2018.
Melbourne finished fifth in the A-League table last season and are affiliated to Premier League champions Manchester City.
In their ranks is midfielder Luke Brattan, who spent some time on loan with Wanderers in 2015 but failed to make a first team appearance under Neil Lennon.
They are also captained by a former Wanderer, Scott Jamie