Having trained with his new team-mates for the first time this Thursday morning, Burke joins up with City initially on a one-month loan deal.
Subject to clearance from The F.A and the Football League, Burke could now be named in Phil Parkinson's match-day squad for this Saturday's Yorkshire derby at Barnsley.
The central defender was named West Ham's Young Player of the Year last season and featured in the Hammers' last five Premier League games for the 2014/2015 campaign, starting four of them.
Burke also played for West Ham in the first round of their Capital One Cup tie at home to Sheffield United in the same season.
Looking to this season, Burke appeared three times for West Ham during their Europa League qualifiers earlier this summer.
The 6' 2'' defender, who turns 19 at the start of next month, has been capped with England at Under 18 and Under 19 level.