The 59-year-old has been replaced at the Stadio San Paolo by former Blues boss Carlo Ancelotti, with Napoli eager to appoint Sarri's successor as soon as possible.
But Chelsea will still have to spend close to £7m to activate the release clause in his current contract if they want Sarri to succeed Antonio Conte in west London.
A report from the Telegraph claims that Sarri is putting pressure on Chelsea in an attempt to force the Premier League side into making a decision over their next manager, and the former Empoli boss is confident that he will be the one to take over this summer.
The report continues to explain how Napoli are also softening on their demand that Sarri's £7m release clause is met - something which is likely linked to their appointment of Ancelotti.
It is claimed that the Italian side would be willing to part ways with their manager if Chelsea defender David Luiz moves the other way as part of the deal.
Luiz has struggled for game time this season and many expect the Brazil international to leave the club this summer, as opposed to the 30-year-old going on a free transfer in 2019.