Tottenham will play Hearts in the play-off round to reach the group stage of the Europa League.
Scottish champions Rangers face a two-leg tie against Slovenian side Maribor while Celtic have been drawn against Swiss side Sion.
Carling Cup winners Birmingham, who were relegated to the Championship, face Portuguese side Nacional and Fulham will play Dnipro of Ukraine.
Beaten FA Cup finalists Stoke City take on FC Thun of Switzerland.
Stoke advanced to the play-off round with a 2-0 aggregate victory over Croatian side Hajduk Spilt.
Fulham also beat a Croatian side 2-0 on aggregate, scoring both goals in Thursday's home leg against RNK Split.
The tie of the round from a British perspective will see Spurs, who finished fifth in the Premier League, travel to Tynecastle to take on Hearts on Thursday, 18 August.
The Scottish Premier League side qualified for the play-off round by beating Hungarian side Paksi 5-2 on aggregate on Thursday.
That second-leg win over Paksi was new manager Paulo Sergio's first game in charge of the club after he replaced Jim Jefferies on Tuesday - and his is now relishing the clash against Harry Redknapp's men.
He said: "No matter who we were drawn against, it was going to be a very difficult tie, but it is a match which I am sure will capture the imagination.
"There is not much travelling involved for the supporters and, of course, there will be the Scotland-England rivalry as well.
"This is a very interesting draw for us, for sure."
Spurs have twice faced Scottish opposition in European competition, beating Rangers 8-4 on aggregate in the Cup-winners' Cup in 1962 and easing aside Aberdeen 5-2 in the Uefa Cup 11 years later.
Rangers, who were knocked out of the Champions League by Swedish side Malmo this week, are away in their first leg against Maribor, a team they beat 6-1 on aggregate in 2001.
Manager Ally McCoist said: "It's a good draw in terms of the place that we are going to. It's accessible and it's not the worst.
"They are a good team and I think they are still sitting at the top of their league. They will be a tough side.
"The club went there seven or eight years ago and got a favourable result over the two legs and we would be hoping for the same.
"It's a tough draw but it's a draw we are pleased with."
Europa League draw for British and Irish teams
Shamrock Rovers v Partizan
Hearts v Tottenham
Maribor v Rangers
Dnipro v Fulham
Sion v Celtic
Nacional v Birmingham
FC Thun v Stoke