The 19-year-old is out of contract at Gigg Lane and will move to Boundary Park with a compensation package likely to be set by a Football League tribunal.
Latics chief executive, Alan Hardy, said: "Dale is a signing for the future.
"He is still a young lad and has only a few first-team games under his belt, so it is likely that he will not figure in the starting line-up at the beginning of the season."
Stephens has played just 12 times for Bury, having gained further experience from loan spells with Droylsden and Hyde United.