Coronavirus cases at Premier League clubs have soared in recent weeks.
The latest round of coronavirus testing revealed 42 positives among the top flight sides.
A club statement on Monday night read: "Manchester United can confirm that our Premier League fixture at Brentford FC, on Tuesday 14 December at 19:30, has been postponed and will be rescheduled in due course.
"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."
The Premier League added in a statement: "With the health of players and staff the priority, and in light of the recent rise in COVID-19 cases across the country, the Premier League has reintroduced Emergency Measures.
"These include protocols such as more frequent testing, wearing face coverings while indoors, observing social distancing and limiting treatment time."
