A host of Premier League sides were understood to be keeping an eye on the youngster's progress, with Bazunu having already broken into the Shamrock first team despite his tender years.
Bazunu forced his way into the team this season, starting in both the League of Ireland and his side's Europa League qualifiers before they were eventually knocked out by Swedish outfit AIK.
The youngster has kept four consecutive clean sheets for Rovers, with a penalty save against Cork City catching the eye of many onlookers.
City are believed to have parted with £420,000 for Bazunu, who has been capped by the Republic of Ireland at U15 level.
Bazunu will likely compete with Daniel Grimshaw and Aro Muric to provide cover for Ederson, with regular back-up stopper Claudio Bravo out long term with an achilles injury.
City academy director Jason Wilcox is believed to have used his strong relationship with Shamrock academy coach Damien Duff in order to push through the move.