Sarri's future as manager has been the subject of much discussion during the end of a campaign in which Chelsea secured a return to the Champions League by finishing third in the Premier League and then won the Europa League, PA reported.
Sarri has been widely linked with Serie A champion Juventus, which is seeking a successor to Massimiliano Allegri.
Reports in Italy said Sarri held talks with Chelsea director Marina Granovskaia on Friday and informed her that he wanted to return to his homeland.
Granovskaia, it is said, promised to talk to club owner Roman Abramovich about his future.
It's understood that Chelsea has no plans to sack Sarri.
Sarri succeeded Antonio Conte as Chelsea manager last summer after spending three years at Napoli.
The 60-year-old enjoyed a tremendous start at Stamford Bridge as Chelsea went unbeaten in 18 games.
But Sarri endured consistent criticism from a significant portion of the club's supporters throughout the campaign and suffered heavy 4-0 and 6-0 defeats at Bournemouth and Manchester City.
Chelsea, however, reached the Carabao Cup final, which it lost on penalties to Manchester City, before finishing the season strongly.
Sarri claimed the first major trophy of his career when Chelsea beat Arsenal 4-1 in Baku on Wednesday to win the Europa League final.
After that game, Sarri said, "You know very well that I love the Premier League, the level of the competition, and I am lucky because I am in Chelsea, one of the best teams in the best championship in the world.
"So at the moment I am happy, but of course I want to know if the club is happy, if we can improve.
"It's normal I think; I have a discussion with the owner after every season with every club."
Chelsea has been linked with their former midfielder Frank Lampard, now the manager of Sky Bet Championship club Derby, should Sarri leave Stamford Bridge.