The ex-England forward has been a free agent since his Anfield contract expired at the end of June.
Sturridge has received a total of 13 offers from clubs in Europe, the United States and Middle East.
Fenerbahce are willing to pay the 29-year-old a £9m salary, but they face competition from fellow Turkish side Besiktas.
Real Betis, Bologna and Marseille are also keen to sign the striker, but MLS outfit DC United could struggle to meet his wage demands.
Sturridge could also move to Qatar, with the former Chelsea man expected to make his choice in the coming days.
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