Serbian international midfielder Kuzmanovic has previously played for Fiorentina in Serie A and has now secured a move back to Italy after a three-and-a-half year stint in Germany.
Stuttgart confirmed the 25-year-old had joined Inter via their official website.
"Zdravko has become a respected player at VfB over the past three-and-a-half years. He now got an offer from Serie A and it has always been one of his wishes to return there. We wish him all the best at Inter," sporting director Fredi Bobic said on the transfer.
Schelotto has signed from Serie A rivals Atalanta and the 23-year-old is delighted to have joined the Nerazzurri.
He told Sky Italia: "Inter are a dream come true and coming here is the best thing possible. I will be available to the coach and contribute to our objectives.
"The road ahead is long and I will show why Inter brought me here. Of course the objective is getting to the preliminary round of the Champions league."
The pair become Inter's third and fourth signings of the January transfer window, following the arrivals of Tommaso Rocchi and Mateo Kovacic.