The Welsh side will look to make use of home advantage, before they head to Montenegro next week for the second leg of the first-round qualifying tie.
Match preview
Last season, TNS collected 92 points from a possible 96 to storm to their third consecutive league title, securing them a spot in Champions League qualifying in the process.
They will now be hoping to progress past the first qualifying round for the first time since beating Kosovan side Feronikeli 74.
Since that success, TNS have experienced two disappointing appearances at this stage of the competition, losing after extra time to Linfield in 2022-23 before slumping to a 5-1 aggregate defeat to Hacken last season.
They have also failed to progress from any of their previous five European ties since beating Kauno Zalgiris in the second round of Europa Conference League qualifying in July 2021.
TNS will also be concerned by the fact they have failed to win any of their previous seven matches in UEFA competition.
Decic are competing in the Champions League for the first time after winning their maiden Montenegrin First League title last season.
They won 20, drew 10 and lost just six of their 36 matches to finish six points clear of second-placed FK Mornar in the standings.
The Montenegrin champions will now look to transfer their domestic success onto the European stage, having failed to do so in their previous two appearances in UEFA competition.
They lost to Drita in the first round of Europa Conference League qualifying in 2021-22, before being beaten by Dinamo Minsk at the same stage of that competition in 2022-23.
After failing to win any of their four UEFA matches, Decic will travel to Wales with aspirations of claiming their first win in European football.
Team News
TNS are likely to line up with Connor Roberts in goal, while Ash Baker, Josh Pask, Danny Davies and Jordan Marshall could all feature in the back four.
Forward Brad Young will provide one of the home side's main goal threats after scoring 29 times in 36 competitive appearances last season.
As for Decic, they are expected to have Igor Nikic in goal, with Jonathan Dresaj, Stefan Milic, Robert Gjelaj and Pjeter Ljuljdjuraj all pushing to appear in the backline.
Bojan Matic, who joined the club in January, is set to lead the line for the visitors after scoring nine goals in 14 appearances in the second half of last season.
The New Saints possible starting lineup:
Roberts; Baker, Pask, Davies, Marshall; Holden, Smith, Clark; Williams, Cieslewicz, Brad Young
Decic possible starting lineup:
Nikic; Dresaj, Milic, Gjelaj, Ljuljdjuraj; Chagas, Brozovic; Puleio, Stijepovic, Gjolaj; Matic
We say: The New Saints 1-0 Decic
With both sides set to play their first game of 2024-25, we think Tuesday's contest could produce a cagey contest as the two teams look to ease themselves into the new campaign.