According to The Sun and The Athletic, Saints boss Ralph Hasenhuttl wants to sign the full-back after missing out on his man in January.
United were unwilling to let Williams leave the club in the January transfer window and it is thought Saints will try to sign him again - either by loan or a full transfer - this summer.
Southampton’s squad depth is significantly lacking at either side of the defensive flank, with Kyle Walker-Peters and Ryan Bertrand the only two senior players that can play that position.
Former captain Bertrand could be set to leave the club in the summer when his contract is due to expire.
Saints’ young full-back pairing of Jake Vokins and Yan Valery appear to not have shown Hasenhuttl enough promise for him to see them as viable back up options to Walker-Peters and Bertrand at this time. The Austrian manager sent both players out on loan in January to Sunderland and Birmingham City respectively.
In Williams, this is a young full-back who can play either side of the defence, showing the kind of versatility this paper-thin Saints squad needs.
The 20-year-old has only played seven minutes of Premier League football for United this season, though has featured for his boyhood club in their domestic and European cup competitions in 2020/21.