Marin has struggled for playing time at Chelsea but still has another two-years left on his current contract.
The 26-year-old winger was signed on an €8 million move from Werder Bremen in 2012 but his Chelsea career never really took off.
Marin has been loaned out to Sevilla, Fiorentina and most recently Anderlecht over the past three seasons.
Trabzonspor want to close the transfer season by signing an additional winger.
Marin has 16 caps for Germany at international level and Bundesliga outfit Augsburg are also reported to be in the race for his signature according to the Daily Mail.
The Black Sea Storm have signed several players this summer including Europa League winner M’bia, Lazio fullback Cavanda and rising Turkish international star Okay Yokuslu.
Trabzonspor are currently managed by Shota Arveladze who replaced Ersun Yanal as head coach over the summer.
The Black Sea based side will not compete in European competition this season.