The 27-year-old Germany international will become a free agent at the end of the season and has been linked with a number of clubs, including Premier League outfits Southampton and Stoke City.
Hunt himself has said that he will not discuss his future until next month and Elchin wants him to considering staying with the River Islanders.
"I am prepared to go to the absolute limit to keep Aaron (Hunt) at Werder Bremen," Eichin told the Weser-Kurier newspaper.
"I will do so because he is very much a key player of our first-team squad.
"At the end, he will have to decide whether he is ready to become a club legend at Werder Bremen or if he wants to try out something new at another club."