Podolski is expected to return to Arsenal in the summer after an unsuccessful loan spell at Inter Milan.
The 29-year-old has made 10 appearances for the Serie A side since joining in January.
The report goes onto state that Arsenal boss Arsene Wenger has had scouts watch Çalhanoğlu in recent months but will face competition for his signature from Manchester City and Liverpool, who have been monitoring his progress in the Bundesliga.
Çalhanoğlu has scored 10 times and provided eight assists in all competitions for the Bundesliga outfit.
The former HSV man has scored seven direct free-kicks in the Bundesliga in his career despite being just 21-years-old.
The Turkish international was also named best young player of the 2013/14 season by leading German football magazine 11 Freunde.