Everton will play Ruzomberok (Slovakia) or Brann (Norway) in the Europa League third qualifying round.
The Toffees, who finished seventh in the Premier League last season, earned the chance to reach the European competition after sixth-placed Manchester United passed their spot down when they won the EFL Cup.
Ronald Koeman's side, with recent acquisition Wayne Rooney in their ranks, will face the two legs before their Premier League campaign kicks off, with all ties set to take place on Thursday, July 27 and August 3.
Aberdeen have been drawn to play Apollon Limassol (Cyprus) or Zaria Balti (Moldova) in their third-round tie, if they beat NK Siroki Brijeg (Bosnia and Herzegovina) in their second qualifying round match next week.
Irish side Shamrock Rovers will face either Kairat Almaty (Kazakhstan) or Skenderbeu (Albania) if they progress against Mlada Boleslav (Czech Republic) in the second qualifying round.
A win for Cork City against Cypriot side AEK Larnaca would see them play Rabotnicki (Macedonia) or Dinamo Minsk (Belarus) in the next round.
Full Europa League third qualifying round draw:
PSV Eindhoven vs Osijek or Luzern
Norrkoping or Trakai vs Shkendija or HJK Helsinki
Krasnodar vs Slovan Bratislava or Lyngby
Sturm Graz or Mladost Podgorica vs Fenerbahce
Panathinaikos vs Gabala or Jagiellonia Białystok
Shamrock Rovers or Mlada Boleslav vs Kairat Almaty or Skenderbeu
Ostersunds or Galatasaray vs Inter Baki or Fola Esch
Gent vs Altach or Dinamo Brest
Zenit vs Trencin or Bnei Yehuda Tel-Aviv
Everton vs Ruzomberok or Brann
Freiburg vs Valur Reykjavik or Domzale
Maccabi Tel-Aviv or KR Reykjavik vs Panionios or Gorica
Dinamo Zagreb vs Vaduz or Odd
Dinamo Bucuresti vs Athletic Bilbao
Valletta or Utrecht vs Haugesund or Lech Poznan
Aberdeen or Siroki Brijeg vs Apollon Limassol or Zaria Balti
Cork City or AEK Larnaca vs Rabotnicki or Dinamo Minsk
Olimpik Donetsk vs PAOK
CSU Craiova vs AC Milan
Irtysh Pavlodar or Crvena zvezda vs Sparta Prague
Zeljeznicar or AIK vs Braga
Arka Gdynia vs Ferencvaros or Midtjylland
Brondby or VPS Vaasa vs Hajduk Split or Levski Sofia
Beitar Jerusalem or Botev Plovdiv vs Maritimo
Liepaja or Suduva vs Sion
Austria Wien vs Progres Niederkorn or AEL Limassol
Bordeaux vs Nomme Kalju or Videoton
Astra Giurgiu or Zire vs Olexandriya
Marseille vs Oostende