The 19-year-old, son of former Manchester City goalkeeper Andy, has put pen to paper on a one-year deal with the option of a further year.
He told Barnsley's official website: "I'm obviously very pleased.
"I was very surprised because one minute I'm signing for Bury and then suddenly I got a call off Willco (Barnsley goalkeeper coach Ian Willcock) saying they wanted to sign me so I'm very pleased about it.
"I think of Ian as a coach, but he's a good lad as well who you can get along with.
"The same is with Ben (Alnwick) and Steeley (Luke Steele), who are two very good goalkeepers.
"I'm here to get better, develop and get fitter. I'm going to get stronger and it can only help that I'm training with better players and coaches who will get the best out of me."