Celtic will face Icelandic side Stjarnan in the second qualifying round of the Champions League.
The Scottish champions begin their campaign at Celtic Park on 14 or 15 July with the second leg a week later on 21 or 22 July.
The Bhoys failed to reach the group stage last season, losing 2-1 over two legs to Maribor in the play-off stage.
West Ham start their Europa League campaign against Andorran side FC Lusitans in the first qualifying round.
The Hammers, who qualified for the competition courtesy of their Fair Play record, will play the first leg at Upton Park on 2 July before travelling to an as yet unconfirmed venue in Andorra on 9 July for the return leg.
This is after the initial schedule was reversed to avoid a fixture clash with Andorra's other representatives, Sant Julia, who were also drawn at home to Denmark's Randers in the first leg.
Should the London club progress, they would come up against either Malta side Birkirkara or Ulisses Yerevan of Armenia.
St Johnstone face Alashkert of Armenia, while fellow Scottish side Aberdeen take on Macedonian side Shkendija.
In the draw for the second round of Europa League qualifying, Inverness were paired with Astra of Romania.
Celtic's Champions League opponents Stjarnan beat Motherwell 5-4 on aggregate in the second qualifying round for the Europa League in 2014-15.
They won their first Icelandic league title in style in October, converting a stoppage-time penalty in their final game to beat FH 2-1 and finish one point ahead of their opponents.
Ronny Deila's side will have to progress through the third qualifying round and a play-off round before they can reach the group stage.
|Selected Champions League qualifying round 2:|
|Celtic v Stjarnan (ICE)|
|BATE (BLR) v Dundalk|
|Skenderbeu (ALB) v Crusaders/Levadia Tallinn (EST)|
|B36 Torshavn (FRO)/The New Saints v Videoton (HUN)|
Manchester United enter the Champions League in the play-off round, having finished fourth in the Premier League last season.
Champions Chelsea, second-placed Manchester City and Arsenal, who came third in the English top-flight, have all already qualified for the group stage.
Southampton are one of the clubs who will join the Europa League in the third qualifying round draw on 17 July, while English Premier League rivals Liverpool and Tottenham Hotspur qualified directly for the group stage.
In Monday's Champions League draw, Welsh Premier League champions The New Saints of Oswestry - one of the seeded teams - were drawn to face B36 Torshavn of the Faroe Islands while Northern Ireland side Crusaders will face Estonian side Levadia Talinn in round one.
|Selected Europa League qualifying:|
|Newtown v Valletta (MLT)|
|Glenavon v Shakhtyor Soligorsk (BLR)|
|NSI Runavík (FRO) v Linfield|
|Progres Niederkorn (LUX) v Shamrock Rovers|
|Cork City v KR Reykjavík (ISL)|
|University College Dublin v F91 Dudelange (LUX)|
|FC Differdange 03 (LUX) v Bala Town|
|KF Shkendija (MKD) v Aberdeen|
|Glentoran v MSK Zilina (SVK)|
|Alashkert FC (ARM) v St Johnstone|
|Skonto FC (LVA) v Saint Patrick's Athletic|
|FC Lusitans (AND) v West Ham|
|AUK Broughton v NK Lokomotiva Zagreb (CRO)|
|Inverness Caledonian Thistle v Astra Giurgiu (ROM)|
Irish club Dundalk face BATE Borisov, the champions of Belarus.
In the Europa League draw, AUK Broughton meet Lokomotiva Zagreb, while fellow Welsh sides Bala Town and Newtown face FC Differdange 03 of Luxembourg and Maltese side Valletta respectively.
Northern's Ireland's Glenavon play Shakhtyor Soligorsk of Belarus, with Glentoran facing Slovak side MSK Zilina.
Shamrock Rovers play Luxembourg's Progres Niedercorn, Linfield tackle NSI Runavik of the Faroe Islands and Saint Patrick's Athletic meet Skonto FC of Latvia.
Cork City will be against KR Reykjavik of Iceland and University College Dublin face F91 Dudelange of Luxembourg.
The Champions League final is on 28 May 2016 at the San Siro in Milan, while the Europa League finale is on 18 May at St Jakob-Park, Basel.