Terry’s contract at Stamford Bridge expires at the end of the season meaning he will be free to discuss terms with other clubs in January.
The Mirror states that Fenerbahce are interested in the former England international and are preparing to table an offer for the 34-year-old at the start of the year.
The experienced defender has been a one-club player over the course of his career. A move away from Chelsea would be the first time he has played abroad or for another club.
Fenerbahçe have become a popular destination for Premier League players following the arrival of Manchester United duo Robin van Persie and Luis Nani last month and ex Blues midfielder Raul Meireles also plays for the Canaries.
The Istanbul giants finished last season as runners-up in the domestic league and are coached by Portuguese manager Vitor Pereira.