The winger has struggled so far at West Ham United.
Arnautovic has struggled to settle at West Ham since his summer switch from Stoke City and Fenerbahce are interested in signing the winger, as reported by Ajansspor.
Although there was a considerable amount of excitement amongst West Ham supporters when he joined from Stoke, Arnautovic's impact has been negligible so far as he has failed to score or provide an assist in nine Premier League appearances.
When at his best and fully focused, Arnautovic can be a major threat to the opposition, as he demonstrated at Stoke, for whom he produced some excellent performances.
However, in what has been a difficult start to the season for West Ham as a unit, the 28-year-old has cut a frustrated figure at times.
It remains to be seen whether West Ham would sanction the sale of Arnautovic so soon after his arrival but doing so would give David Moyes some extra funds to strengthen other areas of his squad in the January transfer window.