Swansea's defeat was their fifth at home this term and their problems grew when Olsson, the only specialist senior left-back at the club, suffered suspected hamstring damage.
Olsson is set to undergo a scan on Wednesday and the fear is that he could be sidelined for some time.
"You're always disappointed when you lose a game," head coach Paul Clement said.
"We actually played better than we have been recently but Manchester United deserved to win the game, no question about that.
"They had the extra bit of quality and from our point of view, we just have to believe a good result and good performance is around the corner.
"We have to show determination to get through this difficult time that we're having."