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The 38-year-old responded in typically irreverent style, saying it was a bad idea for the virus to challenge him.
Milan said that Ibrahimovic, who missed the Europa League match at home to Bodo/Glimt, was placed in quarantine at home and the health authorities had been informed.
The announcement came one day after his team mate Leo Duarte returned a positive test, however Milan said the rest of the squad, who were also tested, were negative.
"I tested negative for Covid yesterday and positive today. No symptoms what so ever. Covid had the courage to challenge me. Bad idea," said Ibrahimovic on Twitter.
Former Barcelona, Juventus, Ajax and Inter Milan forward Ibrahimovic, who will be 39 in October, is the highest-profile Serie A player to test positive since play resumed in June following a three-month stoppage.
One of Europe's most charismatic players, he has scored 14 goals in 22 matches since returning for a second stint at Milan in January, helping revive his team.
He began the new Serie A season in style on Monday by scoring twice in the 2-0 win over Bologna.
His remarkable form led Ibrahimovic to give himself the nickname Benjamin Button after the main character in a 2008 Oscar-winning Hollywood movie about a man who ages in reverse.