Pogba has only just returned to training after being forced into quarantine for two weeks, and is a doubt for United’s Premier League opener against Crystal Palace next Saturday.
The Red Devils began pre-season training later than most of their Premier League rivals after reaching the latter stages of the Europa League.
And with Pogba having returned later than most of his teammates, United’s £89million record signing will have little time to get match fit before the season opener at Old Trafford.
United boss Solskjaer is hopeful the midfielder will be involved, however.
“Paul’s had a couple of training sessions so he’s lost out on a few sessions because of the virus, of course,” the manager Solskjaer told MUTV on Friday.
“But Paul is quick to get fit. Of course, he worked really hard to get back after lockdown last time and played the majority of the games.”
An ankle injury restricted Pogba to just 22 appearances last season.
But 14 of them came after the season resumed following the coronavirus lockdown, and the Frenchman forged a promising partnership with January signing Bruno Fernandes as United surged to a third-placed Premier league finish.
Solskjaer added: “Hopefully, he’ll be ready for next weekend. I am not 100 per cent sure though.”
A weakened United side lost 1-0 to Aston Villa in their pre-season friendly on Saturday.
Villa new boy Ollie Watkins scored the winner, just three days after his £28m signing from Brentford.
Solskjaer gave some of his squad players a chance to impress, with Odion Ighalo leading the line supported by Juan Mata, Andreas Pereira and Jesse Lingard.
Scott McTominay and Fred played at the base of the midfield.
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The only regular in the back four was Victor Lindelof, although his place could come under threat if United add a central defender before the window shuts.
Brandon Williams, Luke Shaw and Diogo Dalot completed the defence, while David de Gea started in goal despite the presence of Dean Henderson – suggesting the Spaniard will remain as United’s No1, for now at least.