This follows positive swab tests for coronavirus on Friday and Saturday.
Essendon's brief statement on Tuesday said the club would "work with relevant medical experts over the coming days to further understand these results".
McKenna went home for the early stages of lockdown before returning Down Under when an AFL resumption was announced.
Prior to Friday's test, the 24-year-old former Red Hands U21 player had tested negative five times since arriving back in Australia.
His positive test resulted in Sunday's game against Melbourne at the MCG being postponed.
McKenna and team-mate James Stewart were sent into a 14-day quarantine following the Irishman's positives tests.
Full-contact training has been banned at all clubs by the AFL in response to McKenna's initial positive tests.
McKenna said in April that he hopes to return home in the next couple of years and play for Tyrone.