The deal includes an obligation for Chelsea to sign the England international, 24, for between £20-25m next summer.
Sancho joined United in a £73m deal from German club Borussia Dortmund in July 2021 but has had a difficult time at Old Trafford.
He made 83 appearances across the three full seasons but had a high-profile falling out with manager Erik ten Hag which led to him training away from the first team.