The club had reported five cases of COVID-19 on Sunday, with Argentina's Ezequiel Garay and France's Eliaquim Mangala both confirming they were among those who had contracted the virus.
"Several further tests for the COVID-19 coronavirus amongst Valencia coaching staff and players have come back positive," the club said in a statement.
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Valencia travelled to Milan last month for a Champions League match against Atalanta, which was played in front of a packed San Siro stadium, before restrictions were placed on sports events in Italy -- the worst-hit country in Europe.
"Despite the strict measures adopted by the club after their Champions League game ... these latest results show the exposure inherent to such matches has caused a positive test rate of around 35%," Valencia added.
"All cases are asymptomatic and those involved are currently isolated at home, receiving medical assessment and carrying out their scheduled training plan."
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The flu-like virus has wiped out the international sporting calendar, with the big five soccer leagues in Europe -- England, Spain, Germany, Italy and France -- as well as the Champions League and Europa League all suspended.
The coronavirus has infected almost 180,000 people and killed over 7,000 worldwide.