The 63-year-old will be leaving his role as the head of sports and development at Lokomotiv Moscow to join the Red Devils in the middle of the season.
Rangnick has agreed to take up the role to manage Manchester United until the end of May as part of the agreement.
The deal is subject to Lokomotiv Moscow agreeing to release the German from his contract, but Manchester United are not anticipating any problems there.
He will not be in the dugout when Manchester United take on Chelsea at Stamford Bridge due to work permit issues.
But the German is expected to take charge of the club next week ahead of their Premier League clash against Arsenal.
Rangnick will also have a consultancy role at Manchester United once his tenure as interim manager ends in May.
Manchester United made it clear to the German that he was their top choice once they started interviewing candidates for the interim role.
It remains to be seen whether he plays a role in helping Manchester United identify a new permanent manager.
