The 20-year-old made 14 Reds appearances in all competitions last season and will relish the opportunity to gain more first-team experience at Carrow Road over the next nine months.
Managed by Daniel Farke, Norwich earned promotion to the Premier League last term by winning the Championship.
Williams will presumably compete for the first-choice left-back spot in Norwich's team, with the Canaries' only other player in that position being Dimitris Giannoulis.
The Manchester-born youth product broke into the United side in the 2019/20 season, making 36 appearances after an injury to Luke Shaw opened up a spot in the side.