Stoke qualified for Europe despite losing in the FA Cup final
Stoke City have been paired with Croatian side Hajduk Split in the third qualifying round of the Europa League.
Fulham will play Slovenia's Domzale or Croatia's RNK Split if they can hold their second round, first leg lead against Northern Ireland's Crusaders.
Hearts were drawn against Tromso of Norway or Hungarian side Paksi.
Dundee United, who trail Slask Wroclaw 1-0
after Thursday's first leg, will face Metalurg Skopje of Macedonia or Lokomotiv Sofia from Bulgaria.
Stoke last played in Europe in the 1974/75 season, when they lost to Dutch giants Ajax in the UEFA Cup.
, telling the club's official website: "It's an occasion that every single one of us are looking forward to.
Tony Pulis was delighted with the draw
"Hajduk Split are a hugely renowned football club with great tradition in European competitions.
"It's also a lovely city for our supporters to go out and visit."
The first leg will be at the Britannia Stadium on Thursday, 28 July. The return match will be at the 35,000 capacity Stadion Poljud a week later.
Welsh Premier League side Llanelli,
who beat former European Cup Winners' Cup champions Dinamo Tbilisi 2-1
in Thursday's first leg, have been drawn against either KR Reykjavik of Iceland or Slovakian team Zilina.
Llanelli manager Andy Legg said his side's display against the Georgians was the best ever by a Welsh team in European competition.
Fellow Welsh side The New Saints face a tough task to reach the third round of qualifying.
They lost their first leg 3-1 at home to Danish side Midtjylland
. If they do progress, they will face Portugal's Vitoria.
Glentoran lost the first leg of their second round qualifier to Vorskla Poltava
of Ukraine 2-0. Their reward for reaching the third round would be a tie against Ireland's Sligo Rovers.
Sligo are managed by the former Wolves and Coventry midfielder Paul Cook.
Europa League third qualifying round draw (UK teams only):
Stoke City v Hajduk Split
Domzale/RNK Split v Crusaders/Fulham
Paksi/Tromso v Hearts
Metalurg Skopje/Lokomotiv Sofia v Slask Wroclaw/Dundee United
KR Reykjavik/Zilina v Llanelli/Dinamo Tbilisi
The New Saints/Midtjylland v Vitoria SC
Glentoran/Vorskla Poltava v Sligo Rovers
Ties to be played on 28 July and 4 August.