When TNS stride out at Stadio Artemio Franchi, they will become the first Cymru Premier club to participate in a UEFA group or league phase, having battled through qualifying; meanwhile, their hosts have reached the final in both of the past two years.
Match preview
Runners-up again last season, Fiorentina came up just short against Olympiacos in the Conference League final, heralding the departure of former coach Vincenzo Italiano and the arrival of new boss Raffaele Palladino.
Italiano also led the Viola to an eighth-placed finish in his final campaign at Stadio Franchi, which was sufficient for another crack at Europe's third-tier club competition this term.
It took a penalty shootout to edge past Puskas Academy in the playoffs, though, and Fiorentina's league form has been mixed since Palladino took the reins.
They had failed to win any of their first four Serie A fixtures for the first time in five years before beating Lazio via two Albert Gudmundsson penalties, and a goalless draw in last week's Derby dell'Arno led to more frustration.
A dull stalemate against Tuscan rivals Empoli has left Fiorentina occupying 13th place in the standings, so Palladino is under pressure to produce a convincing victory on Thursday.
After six draws in eight matches, the Viola would dearly love to post back-to-back wins before the international break, as they host two very different opponents - TNS and then AC Milan - in the space of three days.
With a UEFA coefficient ranking of 162, The New Saints are certainly one of the outsiders to progress from the Conference League's new-look league phase, and they could hardly have kicked off their campaign with a tougher start.
After being eliminated in the early stages of Champions League and then Europa League qualifying, the Welsh side made it through the playoffs by beating Panevezys 3-0 on aggregate, following a comprehensive first-leg win in Lithuania.
Last season, they were knocked out in the Conference League's second qualifying round, but now Craig Harrison's side can look forward to a minimum of eight European fixtures between now and early next year.
The 16-time Cymru Premier champions have met significant opponents in the past - including Manchester City and Liverpool - but a trip to Tuscany will perhaps be the highlight for many of the club's supporters.
Ahead of The New Saints' first encounter with Italian opposition, they recovered from suffering back-to-back league losses for the first time in five years by thumping Newtown 6-1 at the weekend. That result leaves them fifth in the table, albeit with three games in hand.
Fiorentina Europa Conference League form:
D
W
Fiorentina form (all competitions):
D
W
D
L
W
D
The New Saints Europa Conference League form:
W
D
The New Saints form (all competitions):
W
W
W
L
L
W
Team News
Before facing Milan on Sunday evening, Fiorentina are likely to make a number of changes to their starting lineup, with star players such as Moise Kean and David de Gea perhaps being rested.
Raffaele Palladino switched to a 4-2-3-1 formation against Empoli, though it remains to be seen if that continues or the hosts revert to a more familiar back three.
Croatian centre-back Marin Pongracic (thigh) and midfielder Rolando Mandragora (flexor) are both doubts due to injury, but neither has been ruled out; defensive duo Luca Ranieri and Pietro Comuzzo must serve suspensions.
Meanwhile, The New Saints made four changes from their defeat to Bala Town as they crushed Newtown last time out; therefore, Danny Davies, Dan Williams, Rory Holden and Remi Oteh could all keep their places.
Ryan Brobbel and Declan McManus were introduced as substitutes following their return from injury, and after the latter scored his team's fifth with a close-range header he could now be in contention to start; however, fellow forward Jordan Williams will be suspended.
Left-winger Sion Bradley bagged a brace at the weekend and should again provide an attacking threat, though TNS recently lost star striker Brad Young to Saudi Arabian side Al-Orobah.
Fiorentina possible starting lineup:
Terracciano; Kayode, Martinez Quarta, Moreno, Biraghi; Adli, Richardson; Ikone, Beltran, Sottil; Kouame
The New Saints possible starting lineup:
Roberts; Daniels, Bodenham, Davies, Redmond; Brobbel, Smith, D. Williams, Holden, Bradley; Oteh
We say: Fiorentina 3-0 The New Saints
Fiorentina are not likely to field a full-strength team, but their squad depth should be sufficient to avoid a slip-up against TNS.