The Whites have been tipped to bring in a left-back over the course of the transfer window, with a number of options linked with the club.
Fulham left-back Bryan was mooted as a potential target for Leeds earlier in the transfer window, but the club have yet to make any move to take him to Elland Road.
The Cottagers recently signed another left-back in the shape of Antonee Robinson from Wigan Athletic.
And they could be prepared to move Bryan on if offered an acceptable deal.
The former Bristol City star helped Fulham to win promotion to the Premier League last season and is now shaping up for a campaign of top flight football.
Bryan, who will turn 27 years old next month, clocked a total of 49 appearances for Fulham over the course of last season.
He turned out in both his side’s Championship meetings with Leeds last term, allowing the Whites to take a close look at him.
The full-back has another two years on his Fulham deal.
