Pogba has not turned out for United since being forced off as a first-half substitute in the 3-3 draw with Everton on 6 February.
Solskjaer said that the 27-year-old was likely to be unavailable for “a few weeks” following scans on the injury, but now does not expect Pogba to play this month.
“Paul’s recovering. He’s starting the treatment,” the United manager said. “It’ll still be a few weeks, he won’t be playing in February that’s for sure. It’s still a few weeks until we see Paul, definitely.”
Solskjaer has been dealt two fresh injury blows in the form of Edinson Cavani and Donny van de Beek, who are unavailable for Thursday’s Europa League last-32 tie against Real Sociedad.
Anthony Martial and Scott McTominay will travel to Turin for Thursday’s game but both face late fitness tests to determine their involvement.
“Donny and Edinson are injured unfortunately, muscle injuries, so they won’t travel,” Solskjaer said. “Apart from that, we have a couple of doubtful ones, if they’ll be fit enough - Anthony and Scott.”
United’s away leg against Sociedad was moved to Turin due to the Spanish government’s coronavirus travel restrictions on entrants to the country from the UK.
Manchester United travelling squad to face Real Sociedad: De Gea, Henderson, Grant; Bailly, Lindelof, Maguire, Shaw, Telles, Tuanzebe, Wan-Bissaka, Williams; Amad, Fernandes, Fred, James, Mata, Matic, McTominay, Shoretire; Greenwood, Martial, Rashford.
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