Fenerbahce's all-time leading scorer in Europe with 12 goals, Tuncay had been at the Turkish club since 2002 and won three league titles with them.
Boro have been looking to boost their attacking options following Mark Viduka's departure with the side having agreed terms with Arsenal's Jeremie Aliadiere.
Viduka's defection to North-East rivals Newcastle and the release of Malcolm Christie and Danny Graham had left the Teessiders with only two recognised frontmen, Yakubu and Dong-Gook Lee.
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