The Magpies, who were due to travel to Villa in the Premier League on Friday, were forced to close their training ground.
The match is the first Premier League game to be called off due to Covid-19 this season.
A Newcastle statement read: “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.
❌ #NUFC's fixture at Aston Villa on Friday has been postponed following a significant increase in Covid-19 cases at the Magpies’ Training Centre.
A new date and time for the game will be confirmed in due course.
— Newcastle United FC (@NUFC) December 1, 2020
“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.
“The club will not be naming individuals who have contracted Covid-19 and asks that their privacy is respected.”
Newcastle are 13th in the Premier League and are due to host West Brom on December 12.
The Premier League said in a statement: “Newcastle lodged a request with the Premier League to rearrange the game following a significant increase in Covid-19 cases at the club which resulted in Public Health England North East advising that their training facilities remain closed.
“The Premier League board agreed, as a consequence, that the club are unable to train and adequately and safely prepare for their match on Friday.
“The decision has been taken in consultation with Newcastle United and Aston Villa, and following extensive consultation with PHE and Premier League medical advisors, with the health of players and staff the priority.”
The Premier League added that it would rearrange the fixture “in due course”.
A Villa statement added: “We would like to thank both the Premier League board and Newcastle United for their comprehensive and transparent communication during the process which resulted in this decision being reached.
“Everybody at Aston Villa wishes a speedy recovery to all those at Newcastle United affected by the virus.”