Arsenal have been drawn against Napoli, while Chelsea will face Slavia Prague in the Europa League quarter-finals.
Villarreal take on fellow Spanish side Valencia with Portuguese team Benfica playing Germany's Eintracht Frankfurt.
The first and second legs will be played on 11 April and 18 April, with the semi-final draw keeping Arsenal and Chelsea apart should they get that far.
Arsenal boss Unai Emery is chasing a fourth Europa League title as manager after three successive wins at Sevilla.
The Gunners reached the last-eight after overturning a 3-1 deficit to beat French side Rennes 3-0 in the second leg of their last-16 tie.
Should Chelsea win the Europa League, as they did in 2013, it would offer them an alternative route to next season's Champions League.
Maurizio's Sarri's side are currently sixth in the Premier League, three points behind fourth-placed Arsenal, but with a game in hand.
Arsenal first leg switched
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-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
Europa League semi-finals
Napoli or Arsenal v Villarreal or Valencia
Benfica or Eintracht Frankfurt v Slavia Prague or Chelsea