La Real sit top of LaLiga and have only lost twice so far this season in all competitions.
The Spanish club – whose squad includes former United player Adnan Januzaj, as well as ex-Manchester City captain David Silva – will have home advantage in the first leg next February.
Arsenal, meanwhile, will go on the road first in their tie with Benfica, who qualified from Group D behind Rangers.
The Gunners have prospered in Europe this campaign but are struggling badly for form in the Premier League, Sunday's 1-0 defeat to Burnley their fourth in a row at home.
As for their north London neighbors Tottenham, they will go up against Austrian club Wolfsberger when they resume their Europa League campaign in early 2021.
Leicester City will be on the road for the first leg of their tie with Slavia Prague, while Milan will have to get past Red Star Belgrade if they want to remain on course to lift the only European trophy that has eluded the club so far.
Jesse Marsch's Red Bull Salzburg will face Villarreal, who are led by three-time Europa League winning coach Unai Emery.
Steven Gerrard's Rangers go up against Royal Antwerp, Roma face Braga and Napoli will take on Granada.
As for Ajax, who finished third in a Champions League group that included Liverpool and Atalanta, the Dutch club have been drawn with Lille.
The first legs will be played on February 18, with the return fixtures to take place the following week.
Spurs will play their home game on Wednesday February 24, however, so to avoid a clash with Arsenal.
Europa League last-32 draw in full:
Wolfsberger v Tottenham
Dynamo Kiev v Club Brugge
Real Sociedad v Manchester United
Benfica v Arsenal
Red Star Belgrade v Milan
Royal Antwerp v Rangers
Slavia Prague v Leicester City
Salzburg v Villarreal
Braga v Roma
Krasnodar v Dinamo Zagreb
Young Boys v Bayer Leverkusen
Molde v Hoffenheim