The Turkish club has offered Sturridge a two-year contract with an option for an extension, and is willing to include a notable signing-on fee, Sarigul added. Though an agreement has yet to be reached with the Turkish-based agents working on the Englishman's behalf, there's reportedly been significant progress in negotiations.
Sturridge, 29, is a free agent after his Liverpool contract expired following the conclusion of the 2018-19 campaign.
Capped 26 times by the Three Lions, the oft-injured attacker scored twice in 18 Premier League appearances this past season. He chipped in with one tally during the Reds' run to the Champions League crown.
Sturridge was hit with a six-week suspension by the English Football Association earlier this month for breaching betting rules.
Trabzonspor, meanwhile, have already dipped into the market for a former Premier League player this summer, signing ex-Chelsea midfielder John Obi Mikel. A striker remains a priority for the club, according to Sarigul, with Emmanuel Adebayor being considered as well.
After finishing the 2018-19 Super Lig campaign in fourth place, Trabzonspor will compete in the Europa League during the upcoming season.