Juventus defender Daniele Rugani has tested positive for the virus, forcing players and staff of the Italian champions into quarantine.
Madrid players have also been put into isolation after one of the club's basketball players tested positive for Covid-19.
Both matches were due to take place on Tuesday, 17 March.
As it stands, Wednesday's two last 16, second legs between Bayern Munich and Chelsea and Barcelona and Napoli are still set to go ahead behind closed doors.
However, UEFA is to have a crisis meeting on Tuesday with representatives from all 55 national associations and club and player bodies to decide how national leagues, European competitions and Euro 2020 should proceed.