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Silva's return to Spain came via a free transfer, with the midfielder stunning Lazio after opting to sign with Real Sociedad despite previously discussing a deal with the Italian side.
The 34-year-old midfielder has yet to join up with his team-mates for pre-season and will now remain isolated after Monday's positive test.
"Real Sociedad confirms that David Silva has tested positive in the PCR test carried out this morning at the Policlínica Gipuzkoa," the club said in a statement.
"This is the second test that the new Real Sociedad player has taken in the last 72 hours, the first of which took place on Friday in Las Palmas de Gran Canaria and the result was negative.
"David Silva travelled yesterday on a regular flight to Bilbao and arrived in San Sebastián at night.This morning, the second test was carried out, as indicated in the protocol for returning to activity, the result being positive.
"During these first hours in San Sebastian, David Silva has not had the opportunity to go to Zubieta or greet his new colleagues.
"The player is already isolated and has been asymptomatic at all times.The case has been communicated to the competent health authorities."
Silva says that he hassethis sights on winning silverware withReal Sociedadfollowing his switch fromManchester City, as he hopes to help the club challenge the likes of Real Madrid and Barcelona at the top of La Liga.
During his time at Manchester City, Silvawon four Premier League titles, the EFL Cup five times and theFA Cuptwice as he became a key figure during a golden era for the club.
Silva featured 40 times during the most recent campaign, firing six goals in what ended up being his final season in the Premier League.