Anthony Martial, Matteo Darmian, Antonio Valencia, Phil Jones and Mason Greenwood are also doubtful for the trip to London.
"We've not really had a session yet so I don't think there's a lot of positives. I don't know," Solskjaer, who has guided United to eight wins in 10 league games since taking over in December, told a news conference.
Solskjaer has called up midfielders Angel Gomes, Tahith Chong and defender Jimmy Garner and backed the three teenagers to step up to the challenge.
"They have been part of our training for a long time," Solskjaer added. "Angel has been with England Under-17s with Jadon Sancho, Phil Foden and Callum Hudson-Odoi.
"Garner is a Michael Carrick but 20 years younger, Chongy will be taking people on. They are ready."
The Norwegian admitted the decision to rush Lingard back for the Liverpool game was hasty and he will not make a similar mistake with Martial.
"In hindsight you could maybe say Jesse was eager, he ticked all the boxes in training. He'd done the recovery work and rehab but maybe it was a couple of days too early," Solskjaer added.
"Martial isn't going to be ready. Maybe Southampton (on Saturday) but we're not going to risk another one."
United are fifth in the table with 52 points from 27 games.
(Reporting by Shrivathsa Sridhar in Bengaluru; Editing by Ed Osmond)
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