The 30-year-old has been at the Turkish club since the summer after his contract with Liverpool expired, scoring seven goals in 16 appearances.
Trabzonspor are in a four-way tussle for the Turkish title with one automatic Champions League group stage place reserved for the winners and another qualification berth on offer for second place.
It has been reported that a move to Aston Villa could be on the cards for Sturridge this summer but, according to Turkish sports newspaper Fanatik, the striker is keen to stay where he is.
Sturridge's contract expires in the summer of 2021 and then there is an option to extend it by a further 12 months.
The report also claims that Villa tried to make a move for Sturridge in January when they were desperately hunting for a new striker.
The former Chelsea forward rejected the chance to return to England though, staying put in a bid to help Trabzonspor win their first Turkish title since 1984.