The 27 year old stopper will complete his move when his contract expires at Ashton Gate.
"Chris is a player who has been on our wishlist for quite a while now," Pearson told the club website.
"He is desperate to succeed here and we believe he has the ability and the mental strength to do just that."
Weale becomes Pearson's fourth signing since the end of the season, following deals for Jack Hobbs, Wayne Brown and Robbie Neilson.
Bristol City boss Gary Johnson told his club's website: "Chris was never really able to show the fans here what he was about due to injuries.
"For that reason this move is best for all parties."