Ralph Hasenhuttl wants to sign a full-back in the ongoing transfer window and Williams has been Southampton’s top target this month.
The Saints are not alone in their admiration of the 20-year-old as Bayer Leverkusen and Everton have also been linked with an interest in the player.
But Southampton are becoming less confident of signing the defender on loan as it is proving to be difficult to convince Manchester United to let him go.
The south coast club are now looking at other options as they feel signing Williams is likely to be difficult this month.
Manchester United are short of cover at right-back after they sold Timothy Fosu-Mensah to Bayer Leverkusen and Williams is one of the options who plays in that role.
The 20-year-old wants to play more football in the second half of the season and is said to be in favour of a loan move.
But it seems Ole Gunnar Solskjaer wants to keep him around and he is likely to stay at Manchester United until the end of the season at least.
Manchester United did look at signing Atletico Madrid’s Kieran Trippier before the FA handed out a ban to the former Tottenham Hotspur star on betting charges.