Thiago, 32, has yet to feature this season after being injured at the end of last term, but has been tipped to return in the coming days.
With Liverpool having added a number of midfielders to the mix this summer, Thiago could see himself rotated in and out of the side.
Trabzonspor believe that Thiago could be tempted to Turkey and, according to Turkish daily Star, have started efforts to do so.
The Super Lig outfit have signed Nicolas Pepe and club representatives have pressed the button to lure away Thiago from Anfield before the transfer window slams shut in Turkey.
Trabzonspor are looking to bring in Thiago to fortify their midfield options but they could find any potential deal for the Spanish star difficult.
Liverpool are unlikely to want to weaken their midfield options while the transfer window is closed in England.
Thiago was also targeted by clubs in the Middle East this summer, but rejected their overtures in order to stay at Liverpool.
