The 34-year-old Serbian has spent almost a decade at the San Siro, helping the team to five Serie A titles and a UEFA Champions League triumph.
"I don't know if I can find the right words to say goodbye and to thank you properly for everything you have given me, your affection, your trust and your honesty," said Stankovic in an open letter to Inter fans.
"The last 10 years have been the best of my life. I have grown both as a man and, with immense joy, as a footballer.
"My family and I want to thank the Moratti family. I'd like to mention my president in particular: he gave me the opportunity to wear the Inter shirt, which I have been happy and extremely honoured first of all to play in, and then with which I was able to win everything there was to win."
Stankovic, who made 231 appearances for Inter and scored 29 goals, joined the club from Lazio in 2004.
"It's been an honour to wear these colours and they will always remain on my skin. I'll never remove them; nobody can ever remove them," he added.
Stankovic earned 102 caps for Serbia before retiring from international football in 2011.