The former Arsenal and Chelsea attacker joined the Serie A outfit this summer on a permanent deal, after nine years playing in the Premier League.
Giroud has had an instant impact with his new club, as the 34-year-old scored a brace in Milan's 4-1 victory at home against Cagliari on the opening day of the new season.
However, the Daily Mirror reports that the 34-year-old is now set for a spell in isolation after the Rossoneri confirmed that he has tested positive for COVID-19, following a swab test carried out at home.
Giroud's potential absence for Milan's trip to Liverpool on September 15 would leave them short of attacking options, with 39-year-old Zlatan Ibrahimovic currently on the treatment table with a knee injury.
Head coach Stefano Pioli may have to rely on Rafael Leao in a central striker role, while Ante Rebic and Brahim Diaz are other attacking options at his disposal.