Should Cork City progress, they would play the winners of AEK Larnaca (Cyprus) or Lincoln Red Imps (Gibraltar), who famously beat Celtic 1-0 in their Champions League second qualifying round first leg last year.
If Derry City can produce a surprise against Midtjylland, they would encounter Ferencváros (Hungary) or Jelgava (Latvia) in the second round.
Both teams would have a trip away on July 13, followed by a home tie on July 20.
It's the opposite case for Shamrock Rovers, who would welcome Mladá Boleslav of the Czech Republic to Dublin for the first leg if they win their first-round tie.
League of Ireland runaway leaders Cork City will play Levadia Tallinn of Estonia in the Europa League first qualifying round, .
City fans can expect a trip to the 5,000-seater Kadriorg Stadium in the Estonian capital on Thursday week against the team currently second in the Meistriliiga.
City progressed through two rounds of qualifying last year before losing to eventual quarter-finalists Genk.
Cork City FC will face Levadia Tallinn in the @EuropaLeague First Qualifying Round. #CityInEurope pic.twitter.com/mSNuyFftRw
— Cork City FC (@CorkCityFC) June 19, 2017
Derry City will travel to face FC Midtjylland of Denmark in the first leg of their qualifier, while Shamrock Rovers were drawn to play Stjarnan of Iceland.
All Irish sides will be away for their first legs on June 29, before a home leg on July 6.
NIFL Premiership representatives Crusaders will face FK Liepaja of Latvia, while Coleraine and Ballymena United both drew Norwegian opponents.
Coleraine will play FK Haugesund, while Ballymena will face Odds BK.
Teams must progress through four rounds of qualifying to reach the group stages.
The draw for the second qualifying round will be made in Nyom, Switzerland at 1.30pm (Irish time).
First qualifying round draw (matches 29 June & 6 July)
Maccabi Tel-Aviv (ISR) v Tirana (ALB)
Kairat Almaty (KAZ) v Atlantas (LTU)
Beitar Jerusalem (ISR) v Vasas (HUN)
Ordabasy (KAZ) v Široki Brijeg (BIH)
Crvena zvezda (SRB) v Floriana (MLT)
Valletta (MLT) v Folgore (SMR)
Ventspils (LVA) v Valur Reykjavík (ISL)
St Johnstone (SCO) v Trakai (LTU)
Odd (NOR) v Ballymena United (NIR)
Shakhtyor Soligorsk (BLR) v Sūduva (LTU)
Mladost Lučani (SRB) v İnter Bakı (AZE)
Chikhura Sachkhere (GEO) v Altach (AUT)
Fola Esch (LUX) v Milsami Orhei (MDA)
Partizani (ALB) v Botev Plovdiv (BUL)
UE Santa Coloma (AND) v Osijek (CRO)
Zaria Balti (MDA) v FK Sarajevo (BIH)
Bala Town (WAL) v Vaduz (LIE)
VPS Vaasa (FIN) v Olimpija Ljubljana (SVN)
Connah's Quay Nomads (WAL) v HJK Helsinki (FIN)
SJK Seinäjoki (FIN) v KR Reykjavík (ISL)
Shirak (ARM) v Gorica (SVN)
Zirä (AZE) v Differdange (LUX)
Vojvodina (SRB) v Ružomberok (SVK)
Slovan Bratislava (SVK) v Pyunik (ARM)
Rabotnicki (MKD) v Tre Penne (SMR)
Rangers (SCO) v Progrès Niederkorn (LUX)
Domžale (SVN) v Flora Tallinn (EST)
Crusaders (NIR) v Liepāja (LVA)
Nõmme Kalju (EST) v B36 Tórshavn (FRO)
Levadia Tallinn (EST) v
Dacia Chisinau (MDA) v Shkëndija (MKD)
Levski Sofia (BUL) v Sutjeska (MNE)
Irtysh Pavlodar (KAZ) v Dunav Ruse (BUL)
Jagiellonia Białystok (POL) v Dinamo Batumi (GEO)
Željezničar (BIH) v Zeta (MNE)
AEK Larnaca (CYP) v Lincoln (GIB)
Midtjylland (DEN) v
Dinamo Minsk (BLR) v NSÍ Runavík (FRO)
Ferencváros (HUN) v Jelgava (LVA)
Lyngby (DEN) v Bangor City (WAL)
Trenčín (SVK) v Torpedo Kutaisi (GEO)
Lech Poznań (POL) v Pelister (MKD)
Mladost Podgorica (MNE) v Gandzasar-Kapan (ARM)
Videoton (HUN) v Balzan (MLT)
AEL Limassol (CYP) v St Joseph's (GIB)
Skënderbeu (ALB) v Sant Julià (AND)
Haugesund (NOR) v Coleraine (NIR)
Stjarnan (ISL) v
IFK Norrköping (SWE) v Prishtina (KOS)
AIK (SWE) v KÍ Klaksvík (FRO)
Second qualifying round draw (matches 13 & 20 July)
Partizani (ALB) or Botev Plovdiv (BUL) v Beitar Jerusalem (ISR) or Vasas (HUN)
Željezničar (BIH) or Zeta (MNE) v AIK (SWE) or KÍ Klaksvík (FRO)
Haugesund (NOR) or Coleraine (NIR) v Lech Poznań (POL) or Pelister (MKD)
Apollon Limassol (CYP) v Zaria Balti (MDA) or FK Sarajevo (BIH)
AEK Larnaca (CYP) or Lincoln (GIB) v Levadia Tallinn (EST) or
Brøndby (DEN) v VPS Vaasa (FIN) or Olimpija Ljubljana (SVN)
Rabotnicki (MKD) or Tre Penne (SMR) v Dinamo Minsk (BLR) or NSÍ Runavík (FRO)
Kairat Almaty (KAZ) or Atlantas (LTU) v Skënderbeu (ALB) or Sant Julià (AND)
IFK Norrköping (SWE) or Prishtina (KOS) v St Johnstone (SCO) or Trakai (LTU)
Slovan Bratislava (SVK) or Pyunik (ARM) v Lyngby (DEN) or Bangor City (WAL)
Panionios (GRE) v Shirak (ARM) or Gorica (SVN)
Hajduk Split (CRO) v Levski Sofia (BUL) or Sutjeska (MNE)
Stjarnan (ISL) or
v Mladá Boleslav (CZE)Zirä (AZE) or Differdange (LUX) v Astra Giurgiu (ROU)
Nõmme Kalju (EST) or B36 Tórshavn (FRO) v Videoton (HUN) or Balzan (MLT)
Maccabi Tel-Aviv (ISR) or Tirana (ALB) v SJK Seinäjoki (FIN) or KR Reykjavík (ISL)
Chikhura Sachkhere (GEO) or Altach (AUT) v Dinamo Brest (BLR)
Aberdeen (SCO) v Ordabasy (KAZ) or Široki Brijeg (BIH)
Valletta (MLT) or Folgore (SMR) v Utrecht (NED)
Östersund (SWE) v Galatasaray (TUR)
Ferencváros (HUN) or Jelgava (LVA) v Midtjylland (DEN) or
Vojvodina (SRB) or Ružomberok (SVK) v Brann (NOR)
Mladost Lučani (SRB) or İnter Bakı (AZE) v Fola Esch (LUX) or Milsami Orhei (MDA)
Sturm Graz (AUT) v Mladost Podgorica (MNE) or Gandzasar-Kapan (ARM)
Crusaders (NIR) or Liepāja (LVA) v Shakhtyor Soligorsk (BLR) or Sūduva (LTU)
Bala Town (WAL) or Vaduz (LIE) v Odd (NOR) or Ballymena United (NIR)
Dacia Chisinau (MDA) or Shkëndija (MKD) v Connah's Quay Nomads (WAL) or HJK Helsinki (FIN)
Gabala (AZE) v Jagiellonia Białystok (POL) or Dinamo Batumi (GEO)
Ventspils (LVA) or Valur Reykjavík (ISL) v Domžale (SVN) or Flora Tallinn (EST)
Trenčín (SVK) or Torpedo Kutaisi (GEO) v Bnei Yehuda Tel-Aviv (ISR)
Rangers (SCO) or Progrès Niederkorn (LUX) v AEL Limassol (CYP) or St Joseph's (GIB)
Irtysh Pavlodar (KAZ) or Dunav Ruse (BUL) v Crvena zvezda (SRB) or Floriana (MLT)
UE Santa Coloma (AND) or Osijek (CRO) v Luzern (SUI)