Juventus manager Massiliano Allegri and former Barcelona coach Luis Enrique have been the names most heavily linked with the expected vacancy in west London, but ex-France boss Laurent Blanc and now Sarri have also joined those names in the growing rumours.
Apparently talks over a new contract have stalled and he would prefer to hold off making any decision on the subject of his future until after Napoli have faced Juventus in a huge Serie A title decider in Turin later this month.
Four points behind as things stands, if Napoli win that game it would cut the gap at the top to just a single point and perhaps give them the boost they need to go on and complete the job. If they lose, it will almost certainly kill off their challenge and see the title stay with Juve for a seventh consecutive season.
59-year-old Sarri has been with Napoli since 2015 and has been in management since 2000 when he took his first job with lower league Sansovino.
After many years down the league ladder, he gained national attention in Italy in 2014 after guiding Empoli to promotion from Serie B and then exceeding expectations in the top flight to keep them up. That prompted Napoli, his home-town club, to make their move.
Sarri, who is known for smoking during games - which he may struggle to get away with in England, hit headlines for the wrong reasons in January 2016 when he was fined €20,000 and issued a two-game touchline ban for directing homophobic comments towards Roberto Mancini.