The 27-year-old is available on a free transfer this summer after turning down the chance to stay with the Saddlers, with his contract set to expire on June 30.
Oztumer has spent the last two seasons at the Banks's Stadium, scoring 32 goals in 95 appearances.
According to the Sky Sports News, Bolton boss Phil Parkinson is keen to bolster his attacking options after maintaining the club's Championship status on the final day of last season.
The report suggests that Oztumer, who has been dubbed 'the Turkish Messi', is close to joining the Trotters, but Preston North End and Sunderland have also shown an interest in the player.
Oztumer began his career at Charlton Athletic, but was released at the age of 16 and spent three years playing in Turkey before returning to England with Dulwich Hamlet.