The Premier League club have identified the German coach, who is currently the director of sports and development at Lokomotiv Moscow, as a potential short-term successor to the Ole Gunnar Solskjaer.
United sacked the Norwegian on Sunday following a trophy-less three-year spell as boss.
Michael Carrick took charge of the side for Tuesday's Champions League game at Villarreal but the club want to bring in a more experienced figure until the end of the season.
A deal has not yet been finalised with Rangnick and it is understood Carrick will remain in charge for Sunday's game against Chelsea at Stamford Bridge.
Rangnick has a strong reputation and his style has influenced the likes of managers Jurgen Klopp and Thomas Tuchel.
He was also a key figure in the rise of Leipzig through the German lower divisions to the Champions League in the role of sporting director for the club and its Red Bull sister outfit Salzburg.
He has been linked with Premier League clubs in the recent past, including Everton and Newcastle.
United will hope Rangnick can steady the ship at Old Trafford before making a long-term appointment.
Former Tottenham boss Mauricio Pochettino, currently in charge at Paris St Germain, is one of the favourites for the post.
More to follow.