Chelsea manager Sarri has been strongly linked with the Juventus job left vacant after Massimiliano Allegri's departure.
Insigne played under Sarri at Napoli between 2015 and 2018, finishing twice runners-up in Serie A behind eight-time reigning champions Juventus.
"If Sarri went to Juve for us Neapolitans it would surely be a betrayal," Insigne told a press conference with the Italian national team at Coverciano near Florence.
"Choosing Juve will hurt, but we must take note that the coach is a professional and in Naples he gave his all.
"It is his own decision, but I hope in the end he will change his mind."
Under new coach Carlo Ancelotti the club once again finished runners-up behind Juventus, with their last Serie A title coming back in 1990.
"Napoli have not won for a long time, and it hurts us, because we're strong and we've often come so close," said Insigne, who took over the captain's armband in February after Marek Hamsik's departure for China.
"The thing I want most is to raise a trophy with this shirt.
"I have just turned 28, I am neither young nor old, it is the right age to give something more, I have the necessary qualities."
Italy are training before upcoming Euro 2020 qualifiers against Greece and Bosnia-Herzegovina.
"We're building a great group, with the aim of qualifying at all costs for the next Europeans," he added.
"We are Italy, we deserve to participate."