The 1967 European Cup winners made it to the group stages after thumping Cliftonville 5-0 over two legs in the second qualifying round last year, but begin the new campaign with a different coach at the helm.
Ronny Deila is the man charged with filling the boots of former Celtic Park boss Neil Lennon, who won five Scottish titles as a player before guiding the club to three consecutive crowns in the past three seasons.
And Deila will be keen to make a winning start in Iceland this week, with the second leg set to take place at Murrayfield as Celtic's Glasgow home plays host to the Commonwealth Games opening ceremony.
Deila's former club Stromsgodset welcome Steaua Bucharest to Norway for the opening leg of one of the trickier ties of the round, with the Romanians having been crowned kings of Europe back in 1986.
Steaua also featured in the group stage of last year's tournament, but failed to muster a single win as they finished bottom of a pool that also featured Chelsea, Schalke and Basel.
Belarusian champions BATE Borisov's European adventure gets under way against Albania's Skenderbeu Korce on Wednesday, while Dinamo Zagreb tackle Zalgiris Vilnius.
Santa Coloma, HB and Levadia Tallinn have already had to come through one round to reach this stage, beating Banants, Lincoln Red Imps and La Fiorita in the first qualifying round.
The reward for Andorra's Santa Coloma is a two-legged clash with Maccabi Tel Aviv, while HB visit Belgrade to face Partizan after disposing of the first Gibraltarian entrants in the competition's history.
Meanwhile, Estonian league winners Levadia are up against 1992 semi-finalists Sparta Prague.
Ludogorets Razgrad will look to build on an impressive UEFA Europa League campaign last season, but will be wary of their second qualifying round exit in 2012-13 as they take on F91 Dudelange of Luxembourg, while Legia Warsaw get started against St Patrick's Athletic.
Elsewhere in the draw, Maribor are in Bosnia-Herzegovina for the first leg of their clash with Zrinjski Mostar, Slovan Bratislava welcome The New Saints of Wales, Malmo face Ventspils, and Cliftonville are in action against Debrecen.
The round's remaining fixtures see Sheriff Tiraspol host Sutjeska Niksic, HJK come up against Rabotnicki, Dinamo Tbilsi tackle Aktobe, and Qarabag challenge Valletta.
