The 20-year-old centre-back has spent the last six-and-a-half years developing in the Citizens academy, making 89 appearances from Under-18s up to Under-21 level.
Burns celebrated FA Youth Cup success with City in 2020 and was a member of the Under-18 Premier League North title-winning squad in the same year, before winning three successive Premier Leage 2 titles in the last three years with the club's Under-21s side.
The 6ft 5in defender has only made one senior appearance for Pep Guardiola's side, playing the full 90 minutes in a 6-1 EFL Cup win over Wycombe Wanderers in September 2021.
Burns experienced his first loan spell away from the Etihad Stadium when he joined Championship club Swansea City in January 2021, albeit featuring just three times for the Welsh club.
The youngster will now be looking to gain regular first-team football in League One with Stevenage, who secured automatic promotion to the third tier last season after finishing second in League Two.
Burns becomes Steve Evans's 10th signing of the summer transfer window after Harry Anderson, Louis Thompson, Nick Freeman, Nathan Thompson, Alex MacDonald, Dan Butler, Ben Thompson, Taye Ashby-Hammond and Krisztian Hegyi, the latter of whom has joined on loan from West Ham United.
The centre-back could make his Stevenage debut when they travel to Northampton Town for their opening League One fixture of the 2023-24 campaign on Saturday.