A Lions statement on Tuesday confirmed they were exploring a rescheduling of the match or finding different opponents after five positive tests meant it was "impossible" for the Bulls to field a team.
The Lions' meeting with the Sharks in Johannesburg next Wednesday is set to go ahead as scheduled.
However, the Springboks' match with Georgia next week is also in doubt after Nienaber, five management figures and four players - Marvin Orie, Frans Steyn, Handre Pollard and Frans Malherbe - all returned positive tests.
There were also four cases of the virus within the Georgia team. Jurie Roux, CEO of South Africa Rugby, said the news underlined the problems posed by the Delta variant of the virus, which was first identified in India but has since spread to numerous countries.
"These positive results are a setback and have underlined the danger of transmissibility of the Delta variant," he said.
"The Springboks' second Test against Georgia is now in serious doubt, but we will wait for the MAG to consider the data and we will make the final decision tomorrow [Wednesday].
"The priority is to maintain the integrity of the Test series and we will continue to focus on that."
On Monday, South Africa suspended training and placed their squad in isolation after lock Lood de Jager tested positive for coronavirus.
There had been three further confirmed cases of the virus within the squad on June 27, although they were still able to go ahead with their first warm-up match against Georgia, which ended in a 40-9 victory for the Boks.
It was their first match since winning the World Cup in November 2019.
The first Test with the Lions is due to take place on July 24 in Cape Town, before moving to Johannesburg for the second and third matches.