Mann made his Warriors debut against Bath in December 2022 having progressed through the FOSROC Scottish Rugby Academy and impressed playing for Heriot's in the FOSROC Super Series.
The 25-year-old Number 8 was named Mackenzie Construction Most Improved Player of the Season at this year's Awards Dinner, returning to the side against the Scarlets earlier this season having missed nearly two years of rugby with a series of injuries.
Jack Mann said: "It's been a privilege to play for Glasgow Warriors, a club that's helped me grow massively as a player and as a person.
"I'm incredibly grateful to the coaching staff, my teammates, the medical staff who supported me through my rehab, and the supporters for everything over the past couple of years.
"I'm excited for the new challenge ahead with Gloucester and will always look back on my time in Glasgow with pride."
Glasgow Warriors Head Coach Franco Smith said: "Jack has been a committed and hard-working member of our squad, and we've all enjoyed seeing his development over the last couple of seasons.
"He's shown great character, particularly in how he handled his injury and return to play.
"We thank him for his contributions to the club, both on and off the field, and we wish him all the very best for this next step in his career at Gloucester."