Man City are already in the process of offering new terms to Kevin De Bruyne and David Silva and want to reward Jesus for his impact since arriving from Palmeiras for £27m last January.
The 20-year-old has scored 12 goals in 15 starts and his stunning form has made him one of the first names on manager Pep Guardiola’s teamsheet.
Jesus has elevated himself to one of Guardiola’s most significant performers and has exceeded expectations by the speed in which he has settled into the club and the Premier League.
City beat off Barcelona and Real Madrid when they signed him and he agreed a substantial contract when he joined, on a basic salary of around £70,000-a-week.
His current contract runs until 2021 but the Premier League leaders want to acknowledge the improvement he has already made and increase his basic wage and improve his potential bonuses. Jesus has five goals in six City appearances in all competitions so far this season