Stankovic, 34, made only three appearances last year and is now a free agent following his departure from the San Siro giants.
Last month, the Serb insisted he was determined to stay and fight for his place but with no contract extension offered and sides, including Fiorentina and several from the MLS, reportedly holding an interest in him, Stankovic said an emotional goodbye to supporters.
"Dear Nerazzurri fans," he wrote in a statement on the club's official website. "This is your 'Drago' speaking, with a sense of emotion but also with great pleasure.
"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.
"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 made 315 appearances for Inter after joining from Lazio in 2004.
His time with the Nerazzurri brought him five Serie A titles, four Coppa Italia wins and one victory in the Champions League.