The 34-year-old, who joined the LaLiga side this month after leaving Manchester City, is asymptomatic and has already begun a period of self-isolation.
News the midfielder has contracted coronavirus came just hours after he was presented to the media at Sociedad's Reale Arena on Monday.
The club tweeted: "Sociedad informs that @21LVA has tested positive for COVID-19 in the test carried out at Policlínica Gipuzkoa.
ℹ Real Sociedad informs that @21LVA has tested positive for COVID-19 in the test carried out at Policlínica Gipuzkoa. The player is asymptomatic and is self-isolating.#AurreraReala pic.twitter.com/5Rs8gamWsT— Real Sociedad (@RealSociedadEN) August 31, 2020
"The player is asymptomatic and is self-isolating."
Silva becomes the second Real Sociedad player in as many days to contract COVID-19 following the news on Sunday that the winger Mikel Oyarzabal had withdrawn from the Spain squad after testing positive for coronavirus.
Sociedad's LaLiga campaign kicks off with a trip to Real Valladolid on September 13.