Arsenal Women and Glasgow City Women are Britain’s two remaining representatives in the competition, while French club Lyon are the favourites as they bid for their fifth consecutive European title.
The quarter-finals come five months after last-eight ties were originally set to take place and nine months since a ball was last kicked in the competition.
All the remaining games will be one-legged and behind closed doors, after the initial two-legged ties in March were postponed because of the coronavirus pandemic.
Bilbao and San Sebastian are hosting the conclusion of the competition, which will use a similar format to that seen in the men’s Champions League and Europa League.
Bilbao’s San Mames Stadium and San Sebastian’s Anoeta Stadium are the two venues, with the latter staging the final on Sunday, 30 August.
There will not be crowds at the fixtures and all teams are having to adhere to strict safety protocols and Covid-19 testing.
Arsenal, Britain’s only previous women’s European champions in 2007, will meet French side Paris St-Germain on Saturday.
Joe Montemurro’s Gunners overcame Italian club Fiorentina and Slavia Prague of the Czech Republic in the last 32 and last 16 respectively earlier in the tournament.
Lyon are the clear favourites to retain the title, having won the past four finals.
They have been European champions a record six times and reached a further two finals, dominating the competition since 2010.
Source: BBC Sport