The club have stated that they will not reveal the identities of those affected, but a number of players and staff members are self-isolating following positive tests.
The fixture has become the first Premier League match to be postponed since the restart of play during the pandemic.
In a statement, Newcastle said: "Several Newcastle United players and staff members are now self-isolating at home after returning positive test results in recent days and the club's Training Centre site has been temporarily closed in order to contain the spread of the virus.
"With the Magpies' first team currently unable to prepare for the match as a group, Newcastle United lodged a request with the Premier League to postpone the fixture at Villa Park, which has been approved by the Premier League Board.
"The match, which was set to kick off at 8pm (GMT) on Friday evening, will now be rescheduled and a new date and time will be confirmed in due course. Newcastle United would like to thank Aston Villa Football Club for its understanding.
"While the club has continued to ensure strict adherence to these protocols, and despite previously experiencing only a very small number of isolated Covid-19 cases, positive test results have risen sharply in recent days.
"Accordingly, the club is now is working with Public Health England North East (PHE), local health authorities and the Premier League in line with Covid-19 protocols to identify the source of the outbreak and to ensure the risk of further transmission continues to be minimised.
"All players and training ground staff will be re-tested this week and a meeting between the club, Premier League and PHE will take place to assess the test results."
Aston Villa added that they "fully support" the decision to postpone the match.
A date for the rescheduled fixture has not yet been arranged.