"A second round of testing was carried out immediately and produced more positive results today."
The Premier League club's FA Cup third round tie against Liverpool on Friday is now at risk of being postponed.
"First team training ahead of tomorrow’s FA Cup match with Liverpool was cancelled," Villa's statement added.
"Discussions are ongoing between medical representatives of the Club, the Football Association and the Premier League."
Villa are the fourth Premier League club to suffer a Covid-19 outbreak in the last two months with growing concerns over whether the English top-flight can complete the season on schedule.
Three Premier League matches were postponed last week due to outbreaks at Manchester City and Fulham.
On Tuesday, a record high of 40 positive tests were detected among Premier League players and staff in the week between December 28 and January 3.
Southampton's FA Cup trip to Shrewsbury has already been postponed due to a number of positive cases at the League One club.
Wayne Rooney's Derby will fulfil their fixture at Chorley with a youth team after nine positive tests among first-team players and staff.