The striker was released by Liverpool at the end of last season and is still to sign for a new club.
It is believed that the 29-year-old has received few, if any, offers from English clubs so has looked for a new team abroad.
ESPN report that Trabzonspor have made Sturridge an offer worth between £1.84million and £2.77million a year, including bonuses.
The report also says his mother and uncle, who also acts as his agent, have travelled to Istanbul and will arrive in Trabzon on Friday where they will be shown around the region and hold talks.
The Turkish club are thought to be confident of getting Sturridge to sign despite his apparent reluctance to relocate to Trabzon.
He is also believed to have held talks with DC United but they have concerns over his injury history.
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Simon Jordan launched a stinging attack on Daniel Sturridge, describing him as a ‘problem player’ and a ‘divisive influence’ on young players.
The former Crystal Palace chairman Jordan would not consider handing Sturridge a deal if he was still in charge of a Premier League.