Italian rivals AC Milan and Roma meet for a place in the semi-finals, while Portuguese club Benfica take on Ligue 1 side Marseille.
The first legs take place on 11 April, with the second legs on 18 April.
If Liverpool come through their tie, they will face either Benfica or Marseille in the semi-finals, with West Ham meeting AC Milan or Roma if they find a way past Bundesliga leaders Leverkusen.
The semi-finals are scheduled for 2 May and 9 May.
Dublin Arena in the Republic of Ireland hosts this season’s final on Wednesday, 22 May.
Europa League quarter-finals draw
AC Milan v Roma
Liverpool v Atalanta
Bayer Leverkusen v West Ham
Benfica v Marseille
