The Austrian is one of six nominees for the accolade having netted four goals during the calendar month.
Arnautovic's brace helped the Potters to a convincing 2-0 victory over Manchester City before further strikes in the 2-0 win over Manchester United and the 4-3 triumph at Everton.
He also provided the assist for Xherdan Shaqiri's first goal at Goodison Park, playing in all 450 minutes of Stoke's Premier League action.
The five competitors for the award are Dele Ali, Odion Ighalo, Romelu Lukaku, Riyad Mahrez and Mesut Ozil.
Quique Sanchez Flores, Alan Pardew, Claudio Ranieri and Arsene Wenger are the four contenders for the Manager of the Month prize.
Mark Hughes misses out despite earning home wins over both Manchester clubs and another victory at Everton.