The Blues made their move for the England international in the final hours before Friday's transfer deadline.
United have cut their losses on the 24-year-old, who has failed to live up to his price tag ever since a £73 million ($96 million) move from Borussia Dortmund in 2021.
Chelsea will reportedly pay £25 million to make the move permanent in the summer of 2025.
Sancho has not played a league game for United since posting on social media he had been made a "scapegoat" for poor performances just under a year ago.
But he shone during a loan spell back at Dortmund during the second half of last season, helping the German giants reach the Champions League final.
Sancho scored 12 goals in 83 appearances for the Red Devils and won just one trophy - the 2023 League Cup.
He becomes Chelsea's 11th signing of the summer transfer window with the number of players on the club's books having swelled to over 50.
Sancho faces competition from Pedro Neto, Noni Madueke, Mykhailo Mudryk, Cole Palmer, Christopher Nkunku and Joao Felix for a place in Enzo Maresca's side, but said he is excited by Chelsea's project.
"I’m really excited to be here. London is where I grew up and I’m happy to be back," said Sancho in a club statement.
"The manager spoke to me about the project and, for a young player, it's exciting. Hopefully I can bring goals and assists to the Bridge."
