The 27-year-old former Chelsea forward is said to be on the verge of agreeing a move to Turkey, which would be an initial loan with an option to buy.
Abraham's time at Roma appears to be coming to an end, as he spent last season on loan with AC Milan, where he played limited game time.
Throughout his loan spell, Abraham played a total of 45 matches and scored 10 goals, while providing seven assists in 1757 minutes of playing time.
Besiktas are looking to bring in a new striker and are reportedly willing to let go of veteran forward Ciro Immobile to make room for Abraham.
Abraham is seeking guarantees that he will be Besiktas' first choice striker, which is a key factor in his decision to join the club.