The 22-acre site has been acquired on a five-year deal.
The Premier League champions have moved into a new £200m training complex in the city centre at the Etihad Campus.
"The facility at Carrington is out of this world and it was an opportunity that we simply could not turn down," said Bury chairman Stewart Day.
"We are thrilled to have reached this agreement particularly given the keen interest in a number of parties in leasing the site.
"Being able to have a Premier League-standard training facility as our base is a signal of our very real intent to provide our staff and players of all ages with the optimum facilities to support their development and training."
Bury, who currently sit in the final play-off place in League Two, had been training at Lower Gigg, where there was only one pitch.