The 23-year-old winger joined the Turkish giants from Championship outfit Queens Park Rangers in the winter transfer window.
He made 18 appearances in the Turkish Super Lig and did enough to attract the interest of several clubs in Europe this summer.
Celtic and Rangers are interested in signing him and he has also been attracting the attention of Ligue 1 giants Monaco this summer.
A move to Scotland is being mooted, but according to Turkish outlet Enson Haber, Fenerbahce appear to not be keen on letting him go just yet this summer.
The Turkish giants have liked what Osayi-Samuel has added to the squad since joining in January and want him to continue.
Fenerbahce believe he adds value to their squad and are tipped to want to hold on to him in the ongoing transfer window.
It remains to be seen whether any of his suitors make a concrete move for him or whether Osayi-Samuel is ready to leave Fenerbahce just yet.