Spanish world number 16 Garbine Muguruza is a doubt for the US Open because of an ankle injury.
Six of the top 10 players have already pulled out of the women's draw.
One person - not a player - has tested positive for coronavirus at the Western and Southern Open, which begins on Thursday at Flushing Meadows.
Muguruza has withdrawn from that tournament but still hopes to compete in the US Open, which starts on 31 August at the same New York venue.
The person who tested positive for Covid-19 will isolate for at least 10 days and contact tracing has begun to determine if anyone else needs to quarantine.