The Italian coach has been heavily linked with the manager's job at Stamford Bridge, despite the fact that Antonio Conte remains in charge.
After failing to retain the Premier League title, Conte came under pressure at Chelsea, and the latest reports indicate that the Blues are prepared to pay €5m (£4.4m) to release Sarri from his contract at Napoli.
The 59-year-old left the Serie A giants at the end of the season, but he is still under contract, therefore making the move to Chelsea slightly trickier.
According to Sky Sports News, the appointment seems to be edging closer as Sarri's agent has arrived in London to finalise a deal.
Sarri had been in charge of Napoli since 2015.