The striker, who left Liverpool at the end of last season, is set to join his new club until 2022, with the option of an extra year.
Sturridge was released by the European champions when his contract expired in June after six years at Anfield, in which he scored 67 goals in 160 appearances.
In July, the 29-year-old was fined £75,000 and banned for six weeks - four of which were suspended - for breaching betting rules in English football.
Sturridge, who was free to play again from July 31, said it was "extremely disappointing" to hear the Football Association would be appealing against what the governing body considered to be an insufficient punishment.
"I will continue to defend the case and the appeal," he said.
Trabzonspor finished fourth in the Super Lig last season, six points behind champions Galatasaray.
They face AEK in a Europa League play-off first leg on Thursday.