Pellegrini is on the outer of coach Gian Piero Gasperini's plans this season and been encouraged to find himself a new club.
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However, the Italy international has rejected an offer from Trabzonspor.
The Super Lig club were offering a loan-to-buy proposal to Roma, which were keen to strike terms. However, Pellegrini rejected the move as he wants to try and win over Gasperini.
Turkey's transfer window doesn't close until September 12.