United defender Bailly made his first league start since February on Sunday, but had to be taken off after a collision with Mateo Kovacic.
Solskjaer's side are now winless in their past four games in all competitions, and also lost Marcus Rashford to injury in the second half.
But while unable to provide an update on Rashford, Solskjaer confirmed Bailly is unlikely to feature again this term.
"Injuries? Rashy, we strapped him up before the game," Solskjaer told a news conference.
"You could see clearly that his shoulder got worse and worse and he could no longer run freely. Eric, I think it is a bad knee injury.
"Hopefully it is not the worst case. I am sure we won't see him again this season, unfortunately. It just adds to what we got against Manchester City. Anyway, we will be ready against Huddersfield."
United started well at Old Trafford, Romelu Lukaku and Luke Shaw combining to tee up Juan Mata for the opener.
David de Gea's calamitous error enabled Marcos Alonso to equalise before half-time, though, and neither side could make the breakthrough after the restart, and Solskjaer acknowledged that United looked tired in the second half.
"We were tired towards the end of the game, that's for sure," Solskjaer said when asked if his side were not fit enough to carry out his gameplan.
"We have addressed that issue before. We have to take it one step at a time. We depend on everyone being fresh and fit.
"We looked really sharp. Really good. Then for some reason, the second half we didn't come out and play as we did in the first half. There can be many reasons."
The draw leaves United three points behind fourth-placed Chelsea with two games remaining, and Solskjaer suggested any hopes of Champions League qualification are now over.
"If you look at the league table, it is going to be difficult," he added.
"We need to make sure we turn up, win our games and you never know because points are being dropped right left and centre in this race. There's difficult games for all of us. You can't do more than focus on our own performances."