The Senegalese defender was thought to be going to Birmingham when his contract expires next summer, but according to reports Boro have stepped in with a £1 million offer which would see Coly join the Teessiders in January.
It is believed that Steve McClaren sees the right back spot as one of his squads weak areas, despite the impressive performances of Stuart Parnaby in recent weeks.
Coly is available for transfer after having a row with Lens over a new contract. He starred in the summer for Senegal in the World Cup and was one of his sides best players.