The England striker will wait to the end of the transfer window before making his final decision on a future move to Turkey
It can be confirmed they offered the England international a two-year deal worth around €3 million (£2.8m/$3.3m) per season - one of the biggest in the club's history - but are yet to receive a definitive answer on whether Sturridge will accept it.
Despite reported interest from Real Betis, Bologna, Marseille and Besiktas, Trabzonspor are the club to have made the best offer for the 29-year-old free agent.
MLS side D.C. United and clubs in Qatar have also been credited as potential destinations while Sturridge could yet agree a move to the Premier League.
Clubs in the English top flight have until September 2 to name their final squads for the first half of the campaign, and as such Sturridge could still be signed in the next two weeks