A meeting with Michael Laudrup's League Cup winners will be the prize if Hibs can overcome Malmo in their second-round tie.
But the Scottish Cup runners-up must overturn a 2-0 first-leg deficit at Easter Road next week to earn a shot at Swansea.
Swansea midfielder Leon Britton is looking forward to making his European debut and told Sky Sports News: "It means a lot - a lot of us have never played in Europe so it will be a new experience and the fans will look forward to it too.
"Obviously if the game's a 'British derby' then it's not too far to travel but even if it is Sweden then the I think the manager will be happy with that too.
Britton also feels that Swansea's ambitious recruitment drive - including the signing of Jonjo Shelvey from Liverpool and £12m striker Wilfred Bony from Vitesse Arnhem - will stand them in good stead ahead of what could be a busy campaign.
"It's a great sign for the football club that we're able to sign a player for £12m," he added. "It showed last year with Newcastle that the Europa League can affect your league position so it's important to have a big squad."
Elsewhere in the third-round draw, SPL runners-up Motherwell were paired with Russian side Kuban Krasnodar.
And St Johnstone face a tie with Valletta of Malta or Belarussians FC Minsk if they can defend a 1-0 first-leg lead over Rosenborg in Perth next week.
Saints' potential opponents drew 1-1 in the first leg of their second-round tie.
Ireland's Derry City will face Dinamo Minsk or Lokomotiva Zagreb if they overturn a 4-2 deficit to Trabzonspor, while Linfield of Northern Ireland will meet Standard Liege or KR Reykjavik if they defend a 1-0 lead over Xanthi.
The winners of the third-round ties will go into a final qualifying round along with new entrants and the losing teams from the third round of Champions League qualifying.
Selected third-round qualifying ties:
Swansea v Malmo/Hibernian
Dinamo Minsk/Lokomotiva Zagreb v Trabzonspor/Derry City
HNK Rijeka/Prestatyn Town v Olimpija Ljubljana/MSK Zilina
Motherwell v Kuban Krasnodar
Skoda Xanthi/Linfield v KR Reykjavik/Standard Liege
Valletta/FC Minsk v Rosenborg/St Johnstone