The Basque outfit revealed the news on their website on Wednesday following the exams on Tuesday.
"Ahead of starting pre-season preparations, all players, coaching staff and Lezama’s first-team support staff took PCR and serological tests as established LaLiga’s Covid-19 protocol," a statement said.
"Among the results, six have come back positive. The relevant health authorities were notified immediately. The positive cases are now self-isolating in their respective places of residence and will take another PCR test next week."
Athletic did not reveal the identity of those who had tested positive, but Williams confirmed in a tweet that he was one of those affected.
"I'm asymptomatic and luckily I feel good," he wrote. "It's time to stay at home, isolated, until I overcome this virus."
Athletic added that they were following recommendations from LaLiga.
"In accordance with the guidelines set out by LaLiga, the training sessions -the first of which is scheduled for Thursday evening- will start with individual exercises," they said.
"Furthermore, in line with the protocol, all those who tested negative will take new PCR tests on Friday, August 14."
Barcelona, Valencia, Real Betis, Granada, Osasuna, Alaves, Espanyol, Mallorca, Huesca, Las Palmas and Tenerife have also confirmed positives for either players or staff in the past 36 hours.