Albion will be at home to League 2 Grimsby Town in the quarter-finals of the Emirates FA Cup.
The Mariners reached the last eight for the first time in 84 years with a 2-1 victory at Southampton on Wednesday.
It will be the first meeting between the teams since 2004. The sides last met in the FA Cup in 1930 when Albion won the replay 1-0 after a 1-1 draw in the third round.
Quarter-final draw:
Manchester City v Burnley
Manchester United v Fulham
Albion v Grimsby Town
Sheffield United v Blackburn Rovers
Quarter-final ties will be played over the weekend of 18-19 March.