The visitors have not progressed beyond the qualifying rounds since the 2011-12 season, while Copenhagen last did so in the 2016-17 campaign.
Match preview
Having won the Danish Superliga for the first time in three years, Copenhagen boss Jess Thorup will be determined to see his side reap their rewards by reaching the group stage of the Champions League.
The financial income which comes with reaching Europe's elite competition proper is huge for a club of Copenhagen's size, while helping retain them both retain and attract better and more ambitious players in the medium-term.
Having lost three of their first five league games this season, though, it would be fair to say Thorup's side do not head into their crucial two-legged tie against the Turkish champions in the best form, with Friday's 3-1 defeat at home to Randers a particular concern given their usual strength at Parken Stadium.
However, the 52-year-old will undoubtedly have his team fired up for a game of such significance, with an improvement in performance expected to follow.
Trabzonspor, meanwhile, crowned a remarkable season by winning the Turkish Super Lig for the first time in 38 years.
Abdullah Avci had twice guided Istanbul Basaksehir to a runners-up spot in a previous managerial role, but the 59-year-old went one better with Karadeniz Firtinasi, finishing eight points clear of second-placed Fenerbahce in the end.
Former Copenhagen striker Andreas Cornelius led the line magnificently during his debut season for the club, finally coming of age with 15 league goals across the course of the campaign.
The 29-year-old may hold mixed feelings about the prospect of coming up against his local team on Tuesday, but ultimately he will be expected to put professional pride ahead of any personal love and he could play a crucial part in terms of dictating the outcome of the tie.
Copenhagen form (all competitions):
Trabzonspor form (all competitions):
Team News
Copenhagen have a host of injury concerns ahead of Tuesday's home leg, with their main two goalkeepers Mathew Ryan (knee) and Kamil Grabara (facial fracture) both ruled out.
Karl-Johan Johnsson will deputise between the sticks, while Davit Khocholava (shoulder), Nicolai Boilesen (knock), Peter Ankersen (ankle), Mohamed Daramy (knock) and Luther Singh (illness) are all unlikely to feature.
Thorup only utilised club captain Zeca from the bench last time out, but the 33-year-old should come back into midfield in place of Rasmus Falk.
Trabzonspor, meanwhile, will travel without key attacker Edin Visca, who is likely to be out until November with a fractured arm sustained during Friday's 1-0 win over Hatayspor.
Yusuf Erdogan could replace Visca on the right flank, with him and Trezeguet needing to closely support Cornelius in attack.
Veteran midfielder Marek Hamsik is also ruled out due to a calf injury, while Bruno Peres, Serkan Asan and Hakan Yesil are all unavailable.
Copenhagen possible starting lineup:
Johnsson; Diks, Vavro, Khocholava, Kristiansen; Lerager, Zeca, Claesson; Biel, Babacar, Daramy
Trabzonspor possible starting lineup:
Cakir; Larsen, Tokoz, Hugo, Elmali; Omur, Siopis, Bakasetas; Erdogan, Cornelius, Trezeguet
We say: Copenhagen 1-1 Trabzonspor
With both teams missing key players through injury, the quality in general play could be reduced, especially with so much at stake for these sides.
However, Trabzonspor have clearly began the season in better form, and we expect them to avoid defeat and establish themselves as favourites to reach the group stage.