The Red Devils have announced they have closed down their Carrington Training Complex and are in discussions with league bosses over Tuesday's fixture.
The club released a statement confirming the cases on Monday afternoon.
They will now wait for the league's response over whether Tuesday night's clash against Brentford at Old Trafford will go ahead.
"Manchester United can confirm that following PCR test confirmation of yesterday's positive LFT Covid-19 cases among the first team staff and players, the decision has been taken to close down first team operations at the Carrington Training Complex for 24 hours, to minimise risk of any further infection," read the statement.
"Individuals who tested positive are isolating per Premier League protocols.
"Given cancellation of training and disruption to the squad, and with the health of players and staff the priority, the Club is in discussion with the Premier League whether it is safe for Tuesday's fixture against Brentford to continue, both from a Covid infection and player preparation perspective.
"Team and staff travel to London will be deferred pending the result of that discussion."
United are the latest Premier League hit with a outbreak of COVID-19 in their ranks, following a string of similar incidents across the English top flight.
Tottenham Hotpsur were also hit with several confirmed infections which forced them to postpone their fixtures against Rennes in the Europa Conference League and Brighton in the league.
Aston Villa have also suffered an outbreak, with Steven Gerrard's side cancelling a training session at their training ground on Sunday following their trip to Anfield to play Liverpool the previous day.