A total of 18 clubs are already through to the last 32 of Europe’s second-tier competition with six more spots up for grabs on Thursday.
They will be joined by the eight third-placed teams from the UEFA Champions League group stage.
Another English representative, Tottenham Hotspur, are already through along with Spanish clubs Valencia and Sevilla, France’s Lyon and Fiorentina of Italy.
The north London outfit host Russians Anzhi Makhachkala at White Hart Lane with both sides having already qualified from Group K.
Swansea are second in Group A behind Valencia as they travel to face Swiss side St Gallen needing just a point to advance, while Championship outfit Wigan, third in Group D, visit Slovenian outfit Maribor.
Of Swansea’s task, goalkeeper Gerhard Tremmel remarked: “It won’t be easy .. but the whole squad is determined to go to Switzerland and get the result we need to get through.
“Of course, it also depends on how Kuban do in Valencia, but we will only be concentrating on our match.
“A positive result will give us a good feeling and a bit of momentum for the league game at Norwich on Sunday.”
Michael Laudrup’s Swansea were unable to get the point that would have clinched qualification last month as they lost 1-0 at home to Valencia.
For Thursday’s game, Laudrup has said he is reluctant to risk Michu despite the Spaniard’s return from five weeks on the sidelines in Monday’s 1-1 Premier League draw against Hull City, while Ivorian forward Wilfried Bony is recovering from a hamstring problem picked up against Valencia.
“He (Michu) will be tired for a couple of days so we have to think about the rest of the week as we have three games in six days and he cannot play in all three,” said Laudrup.
He added on Bony: “He is a little better, but we have to be careful because we are talking about a muscle injury. With a muscle injury, if you come back one day too early, you could have a setback and be out for four weeks. We don’t want that.”
Last season’s FA Cup winners Wigan know that their fate is out of their hands as they make the journey to Slovenia under new management.
German Uwe Rosler was appointed to replace Owen Coyle, who departed last weekend after a run of three successive defeats left the club from the English northwest in 14th place in the Championship.
Wigan, who lost 2-1 at home to Zulte Waregem last time out, visit Maribor needing to win while also hoping the Belgian side lose to group winners Rubin Kazan.
Elsewhere, Dutch club PSV Eindhoven host Chornomorets Odessa of Ukraine in Group B with all to play for as both have seven points behind already qualified Ludogorets Razgrad of Bulgaria.
German side Freiburg will be looking to seal qualification from Group H alongside Sevilla when they host the Spaniards, although Czech side Slovan Liberec travel to eliminated Estoril also hoping to progress.
Dynamo Kiev host Rapid Vienna with both sides looking for the remaining berth from Group G behind Genk, and Israel’s Maccabi Tel Aviv are on the verge of advancing from Group F.