The two clubs have been locked in negotiations for a number of weeks now, with I Partenopei unwilling to let the Italian leave for anything less than a huge fee - something that has prompted the player to issue the club a warning about.
As reported by Sky, via napolih24.it, the Citizens have now lodged a £53m bid, made up of an initial £47m with £6m in add-ons, which they are hoping will be enough to convince Napoli, who wanted £60m, to let him go.
Jorginho was outstanding throughout 2017/18, gaining plaudits for his ability to dictate the tempo of games in midfield.
His ability to work efficiently in tight spaces and his wonderful range of passing have placed clubs around Europe on high alert, but it appears City are in the driving seat, and there could be significant movement on this deal on Monday.
The 26-year-old, who was born in Brazil, has been with Napoli since 2014 and has played 160 times in all competitions for the club.
He was poached from Hellas Verona, and has developed into one of Serie A's best players over the past couple of seasons, and it looks as though he could now be set to embark on a new challenge in the Premier League.