Zlatan Ibrahimovic requires surgery after sustaining knee ligament damage, but Jose Mourinho is unwilling to give further details about the status of the striker.
Ibrahimovic suffered the injury against Anderlecht in Thursday's Europa League quarter-final at Old Trafford, with Marcos Rojo sustaining similar damage the same match.
Mourinho was reluctant to set a timescale for the pair's recovery - with Ibrahimovic's United contract set to expire at the end of the current campaign - but confirmed both players will go under the knife.
"Everyone is very sad," Mourinho said after United extended their unbeaten league run to 23 matches with a 2-0 win at Burnley on Sunday. "We were thinking about them. They have to be strong.
"Unfortunately they won't be the first or last players to get an important injury.
"Surgery is waiting for them but I don't want to say too much as it's their privacy."
United stopped Burnley from recording a single shot on target in a professional display at Turf Moor that saw Mourinho's men record their fourth straight away league victory thanks to Anthony Martial and Wayne Rooney's goals.
And the Portuguese was pleased with the way his side performed after having to play extra-time against Anderlecht to move into the semi-finals of the Europa League.
"I'm very happy," Mourinho added. "It's not easy to win here and it's even harder to be in control and we were in control for 90 minutes.
"We understood the way they play and we played against their strengths. I think it was a great result that we deserved.
"We wanted Anthony Martial to get behind as he's faster - and he did well. But I can't choose an individual performance as they all did well.
"Paul Pogba and Eric Bailly get a mention after two hours of football at against Anderlecht."