Arnautovic signed a four-year deal with the Potters during the summer, and despite being linked with a return to mainland Europe, the Austrian international has insisted that he is happy at Stoke.
The 24-year-old told The Sentinel: "Yes, I hope to be here for some years. I just arrived, so I cannot say that I want to go in a minute.
"I like it here. I think the people have taken me into the group and helped me to settle in. I am happy to be here and the people are very friendly."
Arnautovic has scored three times in 30 appearances for Mark Hughes's team.