The 31-year-old striker played the full 90 minutes as United slumped to a 2-1 defeat at Swansea City to dent their hopes of finishing in the top four.
Van Persie hurt his ankle when crossing the ball late on as United chased the game, after the Swans had overturned the lead the visitors had taken through Ander Herrera’s opener.
The Dutch striker remained on the pitch, with United having already made all three of their substitutions, and the forward left the stadium on crutches.
Speaking on Sky Sports’ Goals on Sunday, Chelsea boss Mourinho said that United fans should not be fearing the worst about Van Persie, before insisting that he hopes the Dutch forward’s injury is not serious.
Mourinho said: “Maybe in this moment we have this image [of Van Persie on crutches] and it’s just a precaution, because many times now medical departments attack the situation immediately in the stadium.
“Probably this is just a protection to try to attack the injury immediately. I hope it’s like that, I don’t like [seeing] injuries to players.”
The defeat left United fourth in the Premier League table.
MORE: Man United latest news
MORE: Arsenal latest news
MORE: Chelsea latest news
MORE: Liverpool latest news