Arnautovic played 45 minutes of a 1-0 pre-season win over Neuchatel Xamax on Monday, but limped off as Mark Hughes rotated his entire team at half-time.
The 28-year-old joined Stoke from Werder Bremen for £2m in 2013 and signed a new four-year deal in 2016.
Arnautovic, who has played 62 times for his country, has scored 26 goals in 145 appearances for the Potters.
He scored seven times in 35 matches in all competitions in 2016-17.
Stoke City have rejected an offer from West Ham for striker Marko Arnautovic, who has handed in a transfer request.