The defender, 19, leaves the League One side in a bid to secure regular first-team football, and will stay with Cheltenham for the rest of this season.
Robins manager Mark Yates believes that Brundle will be a good acquisition, while he also thanked Bristol City for making the deal possible.
"We needed some more defensive cover just in case. Bristol City have been very good, as has Steve (Cotterill), so we are delighted Mitch has joined us," Yates told the club's official website.
Cheltenham are 13th in League Two.