The right-back has been strongly linked with a move to Craven Cottage over recent weeks, and a bid was made last night.
However, Bardsley will not leave the Stadium of Light today after the offer was rejected.
Bardsley found himself surplus to requirements under Paolo Di Canio earlier this season, and is out of contract in the summer.
However, he has fought his way back into favour under Gus Poyet, and has performed impressively in both the Premier League and in Sunderland’s run to the Capital One Cup final.