The 29-year-old Argentine has struggled to establish himself in the Italian capital, making only sporadic appearances for the Giallorossi this season.
Pellegrini joined Roma from Argentine side Estudiantes in 2015, but has found it tough to break into the team's first XI.
Trabzonspor, who finished third in the Turkish Super Lig last season, are looking to strengthen their midfield ahead of the new campaign.
Pellegrini has yet to comment on the reported offer, but a move to Turkey could provide him with more playing time and a fresh start.
Roma are also said to be open to selling Pellegrini, who has one year left on his contract, as they look to bring in new talent to the club.