The Brazilian currently earns £70,000 per week but, according to the Manchester Evening News and the Telegraph, he is in line for a much improved deal and will extend his current contract, which expires in 2021, for a further 12 months.
The 20-year-old hit the ground running upon his move to England in January, scoring seven times in 10 league appearances, despite missing seven league fixtures through injury.
Having failed to score in City's first two league games of the current campaign, he has since scored four in the last three matches, and also netted as City thumped Feyenoord 4-0 in their Champions League opener last week.
Following a summer that has seen Neymar move to PSG for £200m and Ousmane Dembele join Barcelona in a deal which could be worth £135m, City are understood to be eager to ensure Jesus' contract reflects his status in the team.
Pep Guardiola considers Jesus a pivotal figure in his side, both for the present and in the club's long-term future.
The former Barcelona and Bayern Munich manager is determined to lower his side's average age, in a bid to ensure a period of sustained dominance rather than just short-term success can be achieved. Of Guardiola's 23-man squad, only five are older than 30-years-old, with eight players aged between 26 and 29 and 10 younger than the squad's average age of 25 1/2 years.
According to the Telegraph, however, City will only open negotiations with the Brazilian once they have wrapped up new deals for Kevin De Bruyne and David Silva. The duo's sparkling form has been a major factor in City's strong start to the campaign and the club's hierarchy are looking to tie them down and deter any potential suitors from prising the pair away from Manchester.
De Bruyne is under contract until June 2021 but could reportedly double his £6m-a-year salary once formal talks with the club begin in the next couple of weeks, while City are understood to extend Silva's contract by a further 12 months until 2020, which would see the Spaniard mark a decade at the club.
Fernandinho and Yaya Toure are City's only players whose contracts will expire at the end of the season, after the club released Pablo Zabaleta, Jesus Navas, Gael Clichy and Willy Caballero earlier this summer.