The 24-year-old right back made five Premier League appearances for the West Ham and is understood to be surplus to requirements.
Tony Pulis has identified the right back position an area he needs to strengthen, with Byram a number of targets rumoured to be linked with a move.
Nottingham Forest have also expressed their interest in the former Leeds United defender, who signed on a four-and-a-half year contract in January 2016.
Speaking about the right-back, Nottingham Forest manager and former Boro boss Aitor Karanka said "he is one on the list. He is a West Ham player and we are not in a hurry. We have another few days to try to buy someone and another two or three weeks to try and bring someone in on loan."