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The midfielder could not train at Carrington on Friday and is set to miss the trip to St Mary's.
"Scott [McTominay] and Paul didn't train, they weren't on the grass so it looks very unlikely for the weekend," Solskjaer told a news conference on Friday. "Marcus [Rashford], Victor [Lindelof] and Aaron [Wan-Bissaka] came through the session. They're not 100%.
"Luke [Shaw] has just started his rehab but not out with the team yet. Jesse [Lingard] is back from his isolation period and had a couple of days training. Phil Jones is out until after Christmas sometime."
United head to Southampton, who are fifth in the table after an impressive start to the season, looking to extend their winning run to four. It would be the first time Solskjaer's team have managed four straight wins since July.
"Of course we want that run to continue but we know how well Southampton have done lately -- the last year or so -- but we have a good away record," Solskjaer said. "This is a nice challenge and a good test for the team to see if we learnt our lessons from the games we played last season."
Solskjaer has called last January "a turning point" in his United reign after bringing in Bruno Fernandes from Sporting Lisbon. The next January window is a month away but the Norwegian is not expecting much business.
"It's hard to predict how the financial situation is, if teams need to sell or have got money to buy," he said. "I don't predict a lot of ins and outs.
"The world financially has changed as well and it depends on how injury situations are at teams. I feel very happy with my squad. I have issues every week with an 18-man squad; who do I leave out? We're in a good position."