Arsenal have been drawn against Napoli, while Chelsea take on Slavia Prague in the Europa League quarter-finals.
Villarreal face fellow Spanish side Valencia with Portuguese team Benfica playing Germany's Eintracht Frankfurt.
The first legs will be played on Thursday 11 April with the second legs on 18 April.
Arsenal boss Unai Emery is chasing a fourth Europa League title as manager after winning the competition three successive times at Sevilla.
Arsenal and Chelsea were both drawn at home in the second leg of their ties but both teams are not allowed to play in London on the same or consecutive evenings.
So, the Gunners will now face the Italian side at home in the first leg because Chelsea won last season's FA Cup.
The semi-final draw was also made but it keeps Arsenal and Chelsea apart, meaning they could both meet in the final.
The semi-finals will be played on 2 and 9 May, with the final on Wednesday, 29 May at the Olympic stadium in Baku, Azerbaijan.
Europa League quarter-finals:
Napoli v Arsenal
Villarreal v Valencia
Benfica v Eintracht Frankfurt
Slavia Prague v Chelsea