Arnautovic admitted that he would have had to consider an offer from Dynamo Kiev, who had a bid turned down by Bremen in January, but wants to repay the club's faith in him in the coming weeks.
"Werder have clearly demonstrated that they did not want me to go and want me to stay at the club," Bild quotes the player as saying.
"I have been thinking about the Kiev offer, but I do not look at the money, I look at my career.
"[Turning down the bid] shows the high level of respect for me. Therefore I have no reason to leave. I am happy at Werder and I want to show this in the coming weeks."
Arnautovic has scored five goals in 22 appearances for Bremen this season.