The 20-year-old suffered the injury at the hands of Man City teammate Nicolas Otamendi during Brazil's friendly against Argentina on Friday, and the initial fear was that it might need reconstructive surgery.
However, Jesus has now revealed that he could be back in action within three weeks, leaving him fit in time for the start of Man City's pre-season campaign.
"Yesterday I did a facial resonance, several exams and I have good news! My recovery is going very well and I won't need surgery," he wrote on Facebook.
"In 15 days I will be reassessed but all signs show that I will be back within three weeks! Thanks a lot to everyone that sent me messages, who supported me, sent positive vibes and prayed for me."
Jesus scored seven goals in 10 Premier League appearances following his January arrival at the Etihad Stadium.