The 20-year-old found it hard to get regular playing time at Old Trafford last season, and as a result Ole Gunnar Solskjaer is keen on loaning him out for him to get more opportunities, especially with the return of Diogo Dalot.
The Canaries have reached an agreement with Manchester United for the full-back to stay on loan for the season, but only once Alex Telles fully recovers from his current injury.
Williams has been named as one of United's most highly-rated youngsters to emerge from the academy over recent seasons, and while he has often impressed when given a chance to do so, the form of the likes of Luke Shaw and Aaron Wan-Bissaka, and also Dalot's return and the signing of Telles has limited his opportunities in the squad.
Despite only being 20, he has already made 50 appearances for United, with 21 of them being in the Premier League.
Newcastle were keen on signing him to strengthen their defensive options, while Southampton saw him as the potential replacement for Ryan Bertrand, who left after the expiry of his contract.
However, Norwich have won the race for him, and the deal is set to be completed over the coming days.
Williams made 14 appearances in all competitions last season, and most of his games were in cup competitions or the UEFA Europa League, with Shaw making the left-back position his own in the league, leading to him playing second-fiddle with Wan-Bissaka at right-back.