The Dutch international has seen his career stall since a promising first season for Hamburg, with problems off the pitch affecting his performance on it.
The 25-year-old winger joined Bremen last summer after a nightmare spell at Juventus, where he made just five appearances in just under a year.
His first season back in the Bundesliga was plagued by injury problems - and a speeding offence - but he said that he wants to prove himself to the fans.
"I am out to regain the faith of the authorities and supporters of Werder Bremen," Sky Sports News quotes Elia as saying.
"I want to have impressive performances in the new season. I am not giving up, I am not going to quit the club."
PAOK Saloniki are believed to be interested in Elia.