The 23-year-old Austria international had an indifferent season at the Weserstadion, but expects to show more after the departures of Rosenberg and Pizarro
Arnautovic joined the Bundesliga side in 2010 in a €6.5 million (£5.2m) deal from Twente in 2010, but has experienced a less than impressive season.
The Austria international found himself third in the pecking order this season, having only made 20 appearances and just 10 starts for the club.
However, with Rosenberg and Pizarro set to join other clubs, Arnautovic has stated that he is ready to step up to the role.
"I could help more. Now I am under obligation to show more, I don't feel more pressure than before," the 23-year-old was quoted as saying by Bild.
"Rosenberg, Pizarro, [Tim] Wiese - all gone. But we must move on now, and not give up. We are still good anyway, to get new players. There are other leading players such as Aaron Hunt and Clemens Fritz."
Bremen finished the Bundesliga season in ninth place.