The Hoops will have to negotiate four qualifying rounds if they are to reach the group stages of the competition for a third consecutive campaign.
Armenian league champions Alashkert stand in their way in the first of the four ties, with the first leg scheduled to take place on July 10 or 11, with the return leg a week later on July 17 or 18.
Alashkert are well known to Scottish football fans, having defeated St Johnstone on the away goal rule in the UEFA Europa League first qualifying round in July 2015.
The Yerevan-based outfit, formed in 1990, have played 12 matches in Europe since 2015 - eight in the Champions League and four in the Europa League.
As well as St Johnstone, Alashkert have faced Kairat Almaty of Kazakhstan, Santa Coloma of Andorra on two separate occasions, Dinamo Tbilisi of Georgia and Belarussian side BATE Borisov.
They recorded home wins over St Johnstone (1-0), Kairat (2-1), Santa Coloma (3-0 and 1-0).
Should Celtic get past the Armenians, they will face either Valur Reykjavik or Rosenborg BK.