The 21-year-old England forward completed a medical at United’s Carrington training base last week and the final negotiations of his transfer from Germany have now been completed.
Sancho said: ‘I’ll always be grateful to Dortmund for giving me the opportunity to play first-team football, although I always knew that I would return to England one day.
‘The chance to join Manchester United is a dream come true and I just cannot wait to perform in the Premier League.
United manager Ole Gunnar Solskjaer said: ‘Jadon epitomises the type of player I want to bring to the club, he is a forward player in the best traditions of Manchester United.
‘He will form an integral part of my squad for years to come and we look forward to seeing him blossom. His goals and assists records speak for themselves and he will also bring tremendous pace, flair and creativity to the team.
‘Old Trafford will give him the platform he needs to release his untapped talent and perform at the highest level.
‘For a player of his age, Jadon has already achieved a great deal and showed the courage to go and prove himself abroad. We all look forward to welcoming Jadon to the squad as we prepare for the new campaign.’
Sancho moves back to Manchester just four years after quitting United’s rivals City to move to Germany, where he scored 50 goals in 137 matches for the Bundesliga outfit.