Rashford, 22, suffered a back injury in United's FA Cup third-round replay win against Wolves in January.
It was initially thought Rashford would be out for around six weeks, but United manager Solskjaer said it is not impossible the England forward could now miss the rest of the season and Euro 2020.
"I would hope he's playing this season but it'll just be touch and go towards the end," Solskjaer told British media as he prepared for Europa League last 32 first leg clash Club Brugge.
"So hopefully we can go through and get through this tournament and prolong the season. It takes time. I'm not a doctor, but obviously I was hoping that he would recover quicker than it actually looks like he's going to be out for. So, another few months definitely".