The 24-year-old will join the Robins when his contract expires on July 1 after agreeing terms on Monday.
Weale follows Phil Jevons in swapping Huish Park for Ashton Gate after the striker put pen to paper on a two-year deal last month.
Weale is currently recovering from a cruciate ligament injury and the full terms of his contract are subject to him proving his fitness later in the year, but Johnson is optimistic there will be no problems.
"It's unfortunate he picked up the injury but he's making good progress and we're all positive about his recovery," Johnson told PA Sport. "Chris is a Premiership goalkeeper in the making and a player I've wanted at Bristol City ever since I took the job."
Weale began his professional career at Yeovil in 1999 after coming through the club's youth system and went on to make over 200 appearances for the Glovers.