Stoke's first Europa League tie will see them face Croatian giants Hajduk Split in the third qualifying round.
The teams were paired together in today's draw in Nyon, with the first leg taking place at the Britannia Stadium on July 28 before the FA Cup runners-up travel to Split on August 4.
Fulham, who are 3-1 up on Crusaders after yesterday's first leg, are also likely to face a team from the Croatian city, with RNK Split leading Domzale of Slovenia 2-1 heading into their home match.
Hearts will meet Hungarian side Paksi or Tromso of Norway while a clash with either Metalurg Skopje of Macedonia or Bulgaria's Levski Sofia awaits Dundee United if they can overturn a 1-0 first-leg deficit to Slask Wroclaw.
Hajduk are likely to provide stiff opposition for Stoke and possess an impressive European pedigree.
They have reached the quarter-finals of the European Cup three times, most recently in 1994-95, while they were semi-finalists in the UEFA Cup in 1983-84.
Hajduk have won their domestic league six times, although for the last three years they have had to settle for the runners-up spot behind Dinamo Zagreb.
Fulham's potential opponents RNK finished just behind their city rivals last season to cap a meteoric rise after three successive promotions and are playing in Europe for the first time.
Hearts would expect to be meeting Norwegians Tromso but their second qualifying round tie is very much in a balance after a 1-1 draw in Hungary.
If Llanelli can hold onto their advantage after an impressive 2-1 first-leg win over Dinamo Tbilisi they will meet either KR Reykjavik or MSK Zilina.
Their could be a clash between Northern Ireland and the Republic but Glentoran must overturn a 2-0 deficit to Vorskla Poltava in order to meet Sligo.
Other ties include Ljubljana or Bohemians facing Rudar or Austria Vienna, The New Saints or Midtjylland meeting Vitoria Guimaraes and Karagandy or St Patrick's Athletic taking on Karpaty.
The UEFA Champions League Third Qualifying Round draw is as follows:
Third Qualifying Round
Atletico Madrid v Stromsgodset
Young Boys v TPS/Westerlo
Shakhter/FK Ventspils v Crvena Zvezda
Alania Vladikavkaz v Kecskemeti TE/Aktobe
Floriana/AEK v Mlada Boleslav
Zeljeznicar Sarajevo/FC Sheriff v Khazar Lenkoran/Maccabi Tel Aviv
AZ v Flamurtari/Jablonec
Olimpija Ljubljana/Bohemians v Rudar Pljevlja/FK Austria Vienna
Bursaspor v Gomel
Ferencvaros/Aalesund v FK Suduva/IF Elfsborg
Minsk/Gaziantepspor v Legia Warsaw
Hapoel Tel-Aviv v FC Vaduz/Vojvodina
Olimpi Rustavi/Irtysh v Rennes
Levski Sofia v SK Tirana/FC Spartak Trnava
The New Saints FC/FC Midtjylland v Guimaraes
Dinamo Bucuresti v FC Iskra-Stal/Varazdin
Karpaty Lviv v Shakhtyor Karaganda/St Patricks
Vllaznia Shkoder/FC Thun v Palermo
KR Reykjavik/MSK Zilina v Llanelli/Dinamo Tbilisi
Omonia Nicosia v FC Tauras/ADO Den Haag
SK Liepajas Metalurgs/SV Red Bull Salzburg v FK Senica
Club Brugge v EB Streymur/FK Qarabag
FC 03 Differdange/Levadia Tallinn v Rad Belgrade/Olympiacos Volou
Mainz v KUPS Kuopio/Gaz Metan
UE Sant Julia/Bnei Yehuda v Helsingborgs IF
Stoke v Hajduk Split
FC Gagra/Anorthosis Famagusta v Juvenes/Dogana/Rabotnicki Kometal
Orebro SK/FK Sarajevo v Sparta Prague
Glentoran/Vorskla Poltava v Sligo
Paks/Tromso v Hearts
Metalurg/Lokomotiv Sofia v Slask Wroclaw/Dundee Utd
FH Hafnarfjordur/Nacional v BK Hacken/FC Honka
SV Ried v Brondby
PAOK Salonika v Mika Ashtarak/Valerenga
Domzale/RNK Split v Crusaders/ Fulham
Third round ties to be played 28 July and 4 AugustReuse content