The England international was subbed off in the 73rd minute of the 2-0 win over Manchester City after injuring his left ankle.
Rashford raced back to challenge City winger Riyad Mahrez for the ball and there were fears he could have sustained ligament damage.
However, the 23-year-old underwent a scan on Monday and it was revealed that he has not suffered such extensive damage, according to The Telegraph.
Rashford may still face a spell on the sidelines, though.
His ankle ballooned with swelling bruising and he struggled to put weight on it on Monday.
He will ice it over the next couple of days as the club monitor his progress before determining their next move.
Rashford will surely be a major doubt for the first leg of United’s last-16 Europa League tie with AC Milan at Old Trafford on Thursday.
They face West Ham at home in the Premier League at the weekend.
Solskjaer said after he came off injured on Sunday: ‘When he sprinted 70 yards to save a goal, it showed the attitude of the boy.
‘I think he went over on his ankle but we need to do a scan. I’ve not had time to speak to the doctor yet.’