The daily newspaper states that the Istanbul Giants are in talks with the Brazil international and that they want to sign the 26-year-old in January.
Tottenham were closely linked with Damiao back in 2013 but had a £13 million bid rejected according to the Guardian.
Damiao was recently released by Cruzeiro and is now a free-agent. The powerful striker was the golden boy of Brazilian soccer only a matter of seasons ago.
During the 2011 season Damiao scored 43 goals in 51 games for International attracting interest from across Europe.
Fenerbahce are keen to sign a new striker in the winter transfer window following the departure of Moussa Sow in the summer.
The Yellow Canaries currently have just two strikers, Robin van Persie and Fernandao.
Fenerbahce are currently in first place in the Turkish Super Lig with 37 points in 16 games.