The England international asked to be substituted in the latter stages of his side's 3-1 defeat and appeared to be holding his shoulder in discomfort moments before being withdrawn.
Rashford was forced to pull out of the England squad last month due to a shoulder problem, and Solskjaer has said that the attacker was struggling with the same issue against PSG.
The 23-year-old, who has scored eight times in 15 appearances at club level this term, now faces a battle to be fit for Saturday's Premier League clash with West Ham United.
"It's a niggling shoulder injury. Let's see how quickly he can recover. Hopefully, he will be ready for the West Ham game," Solskjaer told BT Sport.
The forward has only managed two Premier League goals this season but has provided five assists in nine appearances in England's top flight in 2020-21.