According to Turkish Outlet Fanatik, the 27-year-old is about to agree on a move to Turkey that would be a first loan with an option to buy.
Abraham does not seem to have a future at Roma who was on loan last season with AC Milan, where he had a limited playing time.
The striker only played 1757 minutes on 45 games, scored 10 goals and gave seven assists.
Besiktas is assumed that they would like to unload veteran attacker Ciro Immobile to facilitate the deal, with Abraham Warranties looking for he will be their first choice.