Despite this strong interest there are two major stumbling blocks preventing the move.
The first is the need for Allegri to move on Cristiano Ronaldo.
This would free up the wages necessary to bring in a player of Gabriel Jesus' talent. The likelihood of this seems slim though, with few clubs in the current market able to afford Ronaldo's salary demands.
Having said that, Manchester United are rumoured by the Daily Express to have offered their former Portuguese superstar a contract to return to Old Trafford this summer.
Even if the stars were to align, it doesn't appear that Juventus have the money needed to make the transfer happen outright. The deal would require Manchester City to let forward leave on loan with an option to buy at a later date.
This seems unlikely to happen, with Manchester City having let Sergio Agüero's contract run out, and the signing of Harry Kane from Tottenham Hotspur still far from a certainty.
Gabriel Jesus might well be a 'favourite' of Max Allegri, but like much in this transfer window, the ability to get this done relies on a sequence of events that are currently impossible to predict.
Follow us on Twitter for live updates: @City_Xtra