Taking a three-goal lead back home following the first leg in Slovenia, Gala are just 90 minutes away from a tie against either KI Klaksvik or Molde, which would lead on to the group stage.
Match preview
Once wing wizard Kerem Akturkoglu had set them on course for victory with a ninth-minute strike in last week's first leg, Galatasaray never relinquished control of a game that puts them in the driver's seat ahead of Tuesday's return in Istanbul.
Belgian veteran Dries Mertens then extended Gala's lead shortly after the interval, before Halil Dervisoglu stepped off the bench and struck a third goal in stoppage time; effectively sealing the fate of third-round opponents Olimpija.
The Turkish champions' success at Stadion Stozice came amid a summer of fervent activity in the transfer market and followed a nervy second-round win over FK Zalgiris, in which a 1-0 home win edged them through 3-2 on aggregate.
Seeking a return to the Champions League main draw for the first time since 2019, Galatasaray bounced back from a tough 2021-22 campaign to claim their 23rd Super Lig title last term, finishing eight points clear of city rivals Fenerbahce to reclaim top spot after a four-year drought.
Their new domestic season started with a disappointing goalless draw against Kayserispor at the weekend, after a series of underwhelming results in pre-season, but Okan Buruk's side now have a more prestigious prize in their sights.
Should they sign off their place in the playoffs by completing victory over Olimpija, Gala would start favourites to progress past either the champions of Iceland or Norway, and with a star-packed squad featuring stellar names such as Mauro Icardi and Nicolo Zaniolo they are aiming to make an impact on Europe's top club competition this year.
Barring a miraculous comeback, it seems Olimpija are bound for the consolation prize of a spot in the Europa League playoffs later this month, where either Qarabag or HJK Helsinki would await.
Before being crushed 3-0 on home soil last week, the Slovenian club had earned their place in round three of Champions League qualifying with a 3-2 aggregate win over Ludogorets, after eliminating Latvian outsiders Valmiera at the start of the process.
The Dragons lifted the Prva Liga title last season for the first time since 2018, finishing six points ahead of runners-up Celje and 11 clear of outgoing champions Maribor; securing top spot several weeks before the campaign was concluded.
Like Gala, the side overseen by Joao Henriques have already kicked off the new domestic season, starting with a loss in their opener against Koper but bouncing back with a 5-0 win over Rogaska. They then followed last week's European defeat by beating Domzale 2-1 on Saturday, with Marcel Ratnik scoring a second-half winner.
Now, though, Olimpija face one of the most forbidding tasks in European football, as they take on the might of Galatasaray at the Ali Sami Yen.
Galatasaray Champions League form:
D
W
W
Galatasaray form (all competitions):
D
L
D
W
W
D
NK Olimpija Ljubljana Champions League form:
W
W
D
W
L
NK Olimpija Ljubljana form (all competitions):
L
D
W
W
L
W
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Team News
With a tough game against Trabzonspor awaiting at the weekend, Galatasaray could risk making some changes to their starting XI on Tuesday, given their handsome advantage from the first leg.
Cedric Bakambu may therefore make way for Mauro Icardi, while winger Wilfried Zaha is also available and could feature at some stage. Dries Mertens, Nicolo Zaniolo, Kerem Akturkoglu and Baris Alper Yilmaz will also vie for selection in the final third.
Uruguayan midfielder Lucas Torreira has been sidelined by a stomach problem, so Sergio Oliveira competes with Fredrik Midtsjo to join Berkan Kutlu in central midfield, while summer signing Angelino is set to miss out due to injury.
Olimpija, meanwhile, named several fresh faces in their lineup for the league win over Domzale, and some may be retained in Istanbul.
Brazilian forward Pedro Lucas was promoted from the bench on Saturday and will now push Rui Pedro and Admir Bristric for a start up front, but former Derby County midfielder Timi Elsnik is certain to keep his place as captain of the away side.
Galatasaray possible starting lineup:
Muslera; Boey, Nelsson, Bardakci, Karatas; Midtsjo, Kutlu; Yilmaz, Mertens, Akturkoglu; Icardi
NK Olimpija Ljubljana possible starting lineup:
Vidovsek; Silva, Ratnik, Karamatic, Sualehe; Elsnik, Doffo; Seslar, Posavec, Fadida; Pedro