Albion - who announced three or four positive covid tests among their playing staff yesterday - are due to play Wolves tomorrow before the scheduled trip to Man United on Saturday.
Both fixtures are now in jeopardy.
United revealed in a statement that positive Covid tests would require “ongoing surveillance”, leaving the Brentford fixture unplayable in its original scheduling.
The Premier League also announced on Monday a record 42 players and staff tested positive for Covid in the previous week, the highest number since 40 cases were reported in January.
“Manchester United can confirm that our Premier League fixture at Brentford FC has been postponed and will be rescheduled in due course,” read a United statement.
“Following PCR confirmation of positive LFT Covid-19 tests among the first team staff and players, the outbreak requires ongoing surveillance.
“A decision was taken to close first team operations at the Carrington Training Complex for 24 hours to help minimise risk of further infection, and individuals who tested positive are isolating in line with Premier League protocols.
“Given cancellation of training and disruption to the squad, and with the health of players and staff the priority, The Club requested the match to be rearranged.
“The Premier League Board took the decision to postpone based on guidance from medical advisors.
“Manchester United regrets the inconvenience caused to Brentford FC and to the fans of both clubs by Covid-19.”
Brighton were due to face Tottenham last Sunday at the Amex Stadium but that was postponed after an outbreak at the Spurs' training ground.
Speaking yesterday Graham Potter said: "We’re around three or four, but it’s something everyone has got to deal with. Then we’re just making sure it doesn’t go above that number.
"We’ve not spoken about postponing, we’re just monitoring at the moment to make sure you’re not playing guys who have got Covid and are spreading it to the other team. We’ve got to do what we can, keep testing and keep the number as low as possible.
"We’ve had a round [of testing] today and then we will make a decision as to whether or not we have to do another round tomorrow. We’re in that stage now where we don’t want a spread.
"We’re all in the same position. First and foremost the health and safety of the players and everyone around is the most important thing.
"We’ve seen people will miss games because of this, the trick is for it to not be so serious and for it to get too big. Antonio Conte made a good point, you test one day and then test the next and it’s negative, then the following day it’s positive."