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Villa said they had closed their training ground and cancelled today's training session, as a number of players and staff went into isolation ahead of Saturday's (NZ time) game.
"A second round of testing was carried out immediately and produced more positive results today," Villa said in a statement.
"Discussions are ongoing between medical representatives of the club, the Football Association [FA] and the Premier League."
Earlier, the FA said Southampton's FA Cup third round match at home to Shrewsbury Town will not take place as scheduled after the third-tier team reported several COVID-19 cases in their squad.
Shrewsbury told the FA that due to the "significant number" of cases, they did not have enough fit players who were not in self-isolation to fulfil the fixture against the high-flying Premier League side.
"Shrewsbury Town FC provided the FA with details of the medical advice it had received and its communications with local Public Health England and liaised with the FA's COVID Medical Officer for the FA Cup," the FA said in a statement.
"The FA also requested and considered further information regarding Shrewsbury Town's ability to fulfil the fixture."
The FA said there will be a meeting of its Professional Game Board next week to decide on the fate of the fixture.
Championship side Middlesbrough also said preparations for their third round match at Brentford had been affected after a number of their first team players and staff tested positive.
Derby County said their first-team squad and interim manager Wayne Rooney will not play a part in its FA Cup match at sixth-tier Chorley as they were self-isolating after an outbreak of the virus at the club.
But the game is due to go ahead with Derby set to pick a team predominantly made up from the club's under-23 and under-18 squads with youth coaches taking charge.
COVID-19 outbreaks have led to four Premier League matches being postponed and more than 50 games in the lower divisions of the Football League (EFL) being called off this season.