According to reports, the Brazil striker could be sidelined for up to three months, although City have not put a timescale on his recovery.
Jesus went down early in the first half of Monday's 2-0 victory over Bournemouth and received treatment on the field before making way for Sergio Aguero at the Vitality Stadium.
City confirmed he will now undergo further tests in the coming days to determine the extent of the injury.
"We wish Gabriel a speedy recovery and we will keep you updated on his progress via the club's official channels in the coming days," read a statement.
Jesus had scored three goals in three league appearances prior to the trip to the south coast, having arrived in Manchester from Palmeiras last month.
A three-month lay-off would leave Jesus struggling to make a return to first-team action by the end of the season, with City's final Premier League fixture scheduled for May 21 at Watford.