Williams would be an ideal replacement for Ryan Bertrand, who left the club this window on a free transfer for Leicester City.
His ability to provide cover on both sides of the defensive line make him a valuable asset to any team.
Newcastle and Norwich City are also said to be interested in a move for the youngster.
The Saints attempted to sign the versatile full-back last transfer window with no success.
He only managed to start two games for Manchester United last season, with Alex Telles being favoured in the left-back position when Luke Shaw wasn't available.
This year Ole Gunnar Solskjaer looks set to allow several youngsters to go out on loan to develop, with Williams being one of them.
Speaking on a potential move, Alex Crook told talkSPORT: "I told you last week that Brandon Williams was set to join Southampton on loan, I'm now being told that deal could be off due to United's financial demands both in terms of his wages and the fact they want a very substantial loan fee from Southampton.
"The way it was put to me this morning was that if Southampton agreed to United's demands, Williams would effectively become their highest paid player. They're not willing to entertain that at the moment."
United are rumoured to be looking for just over £2million to acquire Williams' services for a season.