Williams, 20, is set to join the south coast side for the duration of the 2021/22 season.
United boss Ole Gunnar Solskjaer opted to keep Williams around his squad last season for squad depth, but the youngster was limited to just four Premier League appearances in total.
Williams is understood to have been frustrated at not being allowed to get regular first-team football over the last 12 months, considering several approach for his services were turned down.
He proved himself as a sturdy replacement for England regular Luke Shaw during his time on the sidelines with injury.
Ralph Hasenhuttl is on the hunt for a new left-back after Ryan Bertrand departed the club for Leicester on a free transfer.
Meanwhile, Saints are in talks with Rovers over a permanent move for prolific Championship hitman Armstrong, who scored 29 goals last season.
Blackburn have already turned one bid down from Southampton for Armstrong, 24, who only has one year remaining on his current deal at Ewood Park.
Rovers have offered their star striker a new and improved deal, although there is no indication at this stage whether he would be willing to accept the terms.