The tie, scheduled to take place over the first weekend of January, is a repeat of last season’s quarter-final.
Wolves memorably triumphed 2-0 on that occasion to book a trip to Wembley, where they suffered defeat to Watford.
Nuno Espirito Santo's team also defeated United at home in the Premier League last term, while the teams played out a 1-1 draw earlier this season.
Villa will travel to Fulham in the FA Cup third round in a repeat of the 2018 Championship play-off final.
The tie will be the first meeting since the Cottagers triumphed 1-0 at Wembley in May last year.
Villa won promotion 12 months later under Dean Smith, while Fulham were relegated after one season back in the top flight. They currently sit third in the Championship.
Albion were, meanwhile, drawn away to Charlton in the third round. It means the Baggies are poised to visit The Valley on consecutive weekends, as they are also scheduled to face the Addicks in the Championship on January 11.