Clubs can not draw teams from the same country and will also be separated into seeds for the first round, decided by UEFA's co-efficient rankings established at the start of the season.
Celtic come into the second qualifying round and must then try to get through the third round and a play-off stage if they are to reach the Champions League group stage for the second year running.
The Dons will play either Ordabasy Shymkent, who finished fourth of in the Kazakhstan Premier League last season, or seventh-placed Bosnian side NK Siroki Brijeg.
May earmarks relief money for London fire victims; death toll at 30
That is despite charitable donations exceeding £3million, and the Prime Minister pledging £5million through an emergency fund. London police said Saturday that the death toll from the inferno at the Grenfell Tower in west London had risen to 58.
Fellow Scottish Premiership side St Johnstone will face FK Trakai of Lithuania in their tie, also at home in the first leg.
The Champions League draw takes place at 11:00 BST, with the Europa League draw to follow at 12:00 BST.
Ballymena United will also face Norwegian opposition in Odd. They were eliminated in the first qualifying round past year after a 5-3 aggregate loss to Estonian side Nomme Kalju.
The Albanian teams will play their first round qualification matches in late June and early July.