The 59-year-old has been linked with Chelsea to replace fellow Italian Antonio Conte, whose future at the English Premier League club remains uncertain.
When asked about Sarri’s future, club president Aurelio De Laurentiis said: “Will Sarri remain? I don’t know, time has run out.”
Fans held up flags with “Sarri, stay with us!” as Napoli ended their record-breaking season with a 2-1 win that condemned Crotone to Serie B football next season.
“In life everything comes to an end,” said Sarri after the match. “Sometimes it’s better to finish stories while they are good.”
“When I arrived, the team had finished 24 points off the top. Now, we are four points behind,” added Sarri who joined Napoli in 2015 from Empoli.
“We have not reached the target, but the journey has been fantastic.”
Lorenzo Insigne orchestrated both Napoli goals for Arkadiusz Milik and José Callejón as the southerners finished on 91 points for the season – four behind champions Juventus.
“Tomorrow I will talk to my family before anyone else,” Sarri continued. “The (club) president needs a response quickly.”
Sarri did rule out going to another Italian club soon.
“I don’t think I could go to another Italian team immediately. The memory of Napoli would be too strong.”