The London club are being linked with a move for the left-back, per Sky Sports.
He was due to sign with Aston Villa, but Bryan has supposedly told Villa that he wants to play in the Premier League.
The 24-year-old was one of the top performers in the Championship last season.
Fulham see him as someone who could hold his own in the Premier League.
The Whites are not done in the transfer market, as they look to try and give manager Slavisa Jokanovic three more players before the window slams shut for Premier League clubs on Thursday.