City have been linked with Jorginho throughout the current transfer window after manager Pep Guardiola identified him as his number one midfield target this summer.
Jorginho is understood to have agreed personal terms with the Premier League champions weeks ago, but discussions are still ongoing between the two clubs as Napoli refuse to lower their asking price.
City are thought to value the Italy international at around £35m, whereas Napoli want closer to £53m despite having already signed a replacement in the form of Fabian Luiz.
The Manchester Evening News reports that Chelsea have now entered the race for Jorginho's signature and could look to hijack the move before City can get it over the line.
Reports that City and Napoli had agreed a £48m deal last week proved to be premature, although City remain Jorginho's preferred destination this summer.
The 26-year-old has been left out of Napoli's pre-season training squad amid the speculation over his future.