Reigning champions Saracens will travel to Leinster in the first quarter-final when this season's Heineken Champions Cup resumes on Saturday, 19 September.
The last eight includes an all-Premiership meeting when Exeter host Northampton on Sunday, 20 September.
Clermont Auvergne will host fellow French side Racing 92, while Ulster will travel to France to face Toulouse.
The draw was in January, with the dates confirmed by European Professional Club Rugby (EPCR) on Thursday.
Ireland's Leinster will be looking to avenge last season's 20-10 final defeat to Sarries.
Saracens, winners of the tournament in three of the last four seasons, will be relegated to the English Championship in 2020-21 after persistent salary cap breaches.
The Challenge Cup quarter-finals will also take place across the same weekend, with Bristol Bears hosting Welsh side Dragons in the first game on Friday, 18 September, while Leicester take on Castres Olympique on the Sunday.
It has not yet been decided whether spectators will be allowed to attend matches, but EPCR says it will adhere to the guidelines at the time.
"All matches will be subject to government guidelines and/or restrictions with EPCR prioritising the health and welfare of players, club staff, match officials, media, supporters and the wider rugby community," said EPCR in a statement after announcing the dates for the fixtures.
Heineken Champions Cup quarter-finals
Saturday, 19 September
Leinster v Saracens (15:00 BST)
Clermont Auvergne v Racing 92 (17:30)
Sunday, 20 September
Toulouse v Ulster (12:30)
Exeter Chiefs v Northampton Saints (17:30)
Challenge Cup quarter-finals
Friday, 18 September
Bristol Bears v Dragons (19:45 BST)
Saturday, 19 September
Bordeaux Begles v Edinburgh (12:30)
Toulon v Scarlets (20:00)
Sunday, 20 September
Leicester Tigers v Castres Olympique (15:00)