Gargano, 28, had fallen out of favour in Naples and after five years at the club has been allowed to leave for the Nerazzurri on loan.
The former Danubio man, who has won 46 caps for his country, believes he would struggle to command a regular place in Napoli's XI, and is poised to join Inter on a one-year deal with the option to buy.
"Dear supporters, I am leaving Napoli after five years," he wrote in an open letter. "It was a difficult decision for me but I want to point out that I asked for the move, and have been fully supported by the club.
"As you know I need to play regularly and the Napoli midfield will be denser than ever this year. There was a serious risk of being left out, so I asked the club to loan me out.
"President Aurelio De Laurentiis has made me happy by sending me to Inter.
"I made my family here in Naples, and my kids were born here. There is a strong bond with the fans and I thank you for the opportunity you gave me, the career you allowed me to have.
"Anything is possible in football and I will not discount returning here at the end of the season. I embrace you all."