The midfielder will miss the trip to Turin for the Europa League round-of-32 first-leg tie with Real Sociedad on Thursday and the return fixture at Old Trafford on Feb. 25.
He has been ruled out of Premier League games against Newcastle and Chelsea and faces a race to prove his fitness ahead of the Manchester derby against City at the Etihad Stadium on March 6.
"Paul is recovering well," Solskjaer told a news conference on Wednesday. "He's starting the treatment but it will still be a few weeks. He won't be playing in February, that's for sure, it's still a few weeks before we see Paul definitely.
"Apart from that, we have a couple of doubtful ones, if they'll be fit enough -- Anthony [Martial] and Scott [McTominay].
"Donny and Edinson are injured unfortunately, muscle injuries, so they won't travel."
Thursday's first leg will be played at Juventus' Allianz Arena after it was moved from Spain amid coronavirus restrictions and Real Sociedad coach Imanol Aguacil said: "We are not going to look for excuses if we play a bad game just because the encounter was played away from Anoeta. I don't like excuses.
"We are very eager, I see my team well, competitive. Despite facing an incredible rival, this is a huge step, a great challenge and we will see if we can progress to the next round. Our only aim is to win tomorrow.
"We are facing a very difficult rival but if we play our game, we will have chances. I hope it's going to be a beautiful game. The two teams can play good football and I hope we are more effective than them. We have maximum respect for Manchester United, for the club's history, their team and their coach."
United will be expected to progress against Real Sociedad, but Solskjaer is wary of the Spanish side who sit fifth in La Liga and have won their last two games.
"We've got to be very good against Sociedad," Solskjaer said.
"When the draw was made it was probably the hardest you could pick from the unseeded group. They were top of La Liga for a while and they're still in the hunt for Champions League for next year.
"They've got some top players, some very experienced players, some quality players and some young very good ones and we have to play at a high level to get a result. The Spanish league is also a quality one."
Mason Greenwood is part of the group which made the trip to Italy on Wednesday following the news on Tuesday that the 19-year-old has signed a new long-term contract at United.
Greenwood is getting back to his best after scoring 17 goals last season and says he is learning to cope with the extra attention after his breakthrough campaign.
"It's just a challenge after you've played for the first season not many of the players know what you're going to do, in the second season you've really got to have a few more tricks up your sleeve," he said.
"I feel like I'm getting back up to where I was last season. I've been here since the age of seven and it's the place I want to be. Hopefully I can get back up to the standards I was up to last season and I'm working hard with all the coaches so hopefully I'll be back there soon."