The seven-times European champions have missed out on the group stage only once in the last 11 seasons and failure to qualify this term could quickly turn the heat back on coach Massimiliano Allegri, who clung on to his job during the summer.
Arsenal, aiming to reach the group stage for the 16th season in a row, visit Fenerbahce in another of the 10 ties while Huub Stevens returns to his old club as he leads PAOK Salonica against Schalke 04 and Olympique Lyonnais host Real Sociedad in a France-Spain contest.
PAOK are one of five clubs, alongside Ludgorets Razgrad, Shakhter Karagandy, Pacos de Ferreira and Austria Vienna, who are attempting to reach the group stage for the first time.
If Shakhter overcome former European champions Celtic, they would also become the first team from Kazakhstan to reach the group stage since the Central Asian nation joined UEFA from the Asian Football Confederation in 2002.
Dutch clubs are no longer the force they were in European competition but even so, Milan, third in Serie A last season, will not be relishing their trip to 1988 champions PSV, who have already played five games this season and won them all.
"It will be tough for Milan because PSV are a young team, who run a lot and want to play on the attack," former Netherlands midfielder Mark van Bommel, who played for both clubs during his bruising career, told Milan's website (www.acmilan.com).
"PSV are very good at going forward but they have a very young defence. It will be a very interesting match because who knows what can happen."
PSV have sold several key players from last season but have been boosted by the emergence of 17-year-old Belgian Zakaria Bakkali, who became the youngest player to score a hat-trick in the Eredivisie in this month's 5-0 win over NEC Nijmegen.
Arsenal travel to Istanbul for Wednesday's match with their confidence dented and questions over manager Arsene Wenger's transfer policy after a 3-1 home defeat by Aston Villa in the Premier League on Saturday.
Fenerbahce are surrounded by uncertainty after they were handed a two-year European ban in June over a domestic match-fixing scandal in 2011 but were reinstated pending an appeal at the Court of Arbitration for Sport (CAS).
CAS are due to announce the result before Aug. 28, the day after the second leg.
Ukraine's Metalist Kharkiv, meanwhile, were expelled last week by UEFA in connection with a domestic match-fixing case dating back to 2008 and replaced by Greek runners-up PAOK, who visit Schalke on Wednesday.
Unlike with Fenerbahce, CAS refused to grant Metalist a temporary reprieve after they also appealed.
PAOK coach Stevens had two stints in charge of the Gelsenkirchen-based club, the most recent ending only eight months ago when he was fired.
Celtic, champions in 1967, set off immediately after Saturday's Scottish league match at Aberdeen for a 5,500-km (3417 miles) journey which took them across five time zones for Tuesday's game against Shakhter.
Tuesday's match will be played in Astana, around 400 km from Shakter's own stadium which did not meet UEFA requirements for European competition.
Real Sociedad's match at Olympique on Tuesday will be their first in Europe since the 2003/04 season and coincidentally will be against the same opponents.
Olympique, meanwhile, are looking to return to the Champions League group stage after their run of 12 successive participations ended last season.
In other ties, Pacos de Ferreira, who had never finished higher than sixth in the Portuguese league before last season's impressive third place, host Zenit St Petersburg on Tuesday, hoping the Russians fail to notch a first-ever win in Portugal.
Former European Cup semi-finalists Austria Vienna, aiming to become the first team from their country to qualify for seven years, visit Croatian champions Dinamo Zagreb on Wednesday.
Steaua Bucharest, champions in 1986, host Legia Warsaw the same evening when Bulgarian champions Ludgorets Razgrad entertain Switzerland's FC Basel.