Jelavic is expected to part ways with the Tigers in the summer after the club was relegated to the Championship.
Hull City signed Jelavic for £7 million from Everton in 2014 and he still has two years left on his contract.
Trabzonspor are set to offer Jelavic terms that would match his reported £55,000-a-week wages and offer him the chance to compete in the title race next season.
The 29-year-old scored eight goals and provided one assist in 29 appearances for Hull City last season. Jelavic has a total of 13 goals and two assists for the Tigers since joining from Everton.
Trabzonspor are managed by Ersun Yanal who led Fenerbahçe to the league title the season before last.
The Black Sea Storm finished last season in fifth place but are expected to invest heavily and mount a title challenge next season.