Ulster will face English Premiership side Harlequins in the last 16 of the European Challenge Cup at the Stoop on Sunday 4 April, (20:00 BST kick-off).
The winners of the match will face the Dragons or Northampton Saints away in the quarter-finals the following week.
Connacht play Leicester Tigers at 20:00 BST on Saturday 3 April.
In the Champions Cup, Leinster host Toulon at 17:30 on Friday 2 April with Munster v Toulouse scheduled for at 15:00 on Saturday 3 April.
Ulster defeated Harlequins in both European Champions Cup encounters between the sides in December 2019.
It included a memorable 25-24 last-gasp victory at Kingspan Stadium.
Dan McFarland's team dropped into the Challenge Cup having lost their two Champions Cup group games.
Should Leinster overcome Toulon, they will be away to the winners of the Exeter Chiefs-Lyon encounter, meaning a possible quarter-final trip to Sandy Park to take on the reigning champions.
Munster, meanwhile, will be rewarded with an away quarter-final against the winners of Wasps v Clermont Auvergne.