The 19-year-old versatile defender has penned an 18-month deal at Montgomery Waters Meadow after leaving boyhood club City, where he has been since aged 10.
Ogbeta, an England youth international, is described as a ball-playing defender who can play both left-back, at centre-half and also in midfield.
The teenager, who hails from Salford, has been a Premier League 2 regular for City's under-23s. He did not make a first team appearance for Pep Guardiola's side.
He did feature against Shrewsbury for City's youth side in the 2019/20 EFL Trophy group tie at Montgomery Waters Meadow, which the youthful visitors edged on penalties.
He has played in EFL Trophy fixtures this season too, and was reported to be interesting German clubs in recent weeks.
Ogbeta arrives as a replacement for the departed Charlie Daniels, who left at the end of a short-term deal and has joined League One rivals Portsmouth.
His arrival comes after a move for Luton left-back Brendan Galloway broke down at the final stages.
Scott Golbourne was forced to play at left wing-back having not regained full fitness from a serious rib injury and right-back Donald Love was used in the unfamiliar role at Sunderland on Saturday.
Ogbeta was named in England's under-20 squad last November alongside top City youngsters Tommy Doyle and Taylor Harwood-Bellis. The youngster was out of contract at the Etihad Stadium this summer.
Town are believed to be chasing striking reinforcement and a right wing-back in the remainder of the window, which shuts next Monday.