The 30-year-old Bart-Williams was training with Leeds for a month hoping to impress Kevin Blackwell, but the club have not offered him a deal and the midfielder could now join former club Nottingham Forest, also on trial.
Meanwhile, Darlington may move for Craig Hignett if his month-by-month contract with United is not extended when it expires next week.
"We're first in the queue if Leeds don't extend his deal." Quakers boss David Hodgson told the Evening Gazette.
"He has a lot to offer and we would love to have him."