Grant, 19, moves to the Potters on a three-year contract.
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Grant, the son of former Potters goalkeeper Lee, spent last season on loan at National League North side Buxton, where he made 36 appearances.
"True has shown his quality at Manchester City and during his loan spell at Buxton, and he is regarded as one of the most promising young goalkeepers in the country," Stoke sporting director Jonathan Walters told the club website.
"Darren Behcet was keen to bring him in as part of a strong goalkeeping unit with Viktor Johansson, Jack Bonham and Frank Fielding, with Tommy Simkin also excelling whilst out on loan.
"His dad, Lee, was a hugely respected player here and a fantastic person too."