United will also be without Pogba who is recovering from an ankle and foot injury after playing just twice since August.
The Mirror claim early indications suggest the injury isn't serious but De Gea will be fully assessed by Man Utd's medical team and undergo a scan today before the full extent of the injury is know.
France midfielder Pogba has missed five of United's last seven matches with an ankle injury, and after spending part of the global break recuperating in Dubai, Solskjaer said the 26-year-old is still not ready for first-team action.
De Gea has not missed a league game since May 2018. "I think he'll be out".
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De Gea's injury came in innocuous circumstances - he immediately grabbed the back of his leg after clearing a backpass from his own player.
Sergio Romero would likely deputise in his absence.
Watch Manchester United vs Liverpool live on Super Sunday from 3pm on Sky Sports Premier League; kick-off 4.30pm. "It certainly looked like it anyway judging on last night so it's one of those things".
"We had a scan after the Arsenal game and he needed a few weeks rest in a boot".
"Hopefully Aaron and Anthony will last the training this week and be available for selection", said Solskjaer.
Liverpool have surged to the top of the table after a flawless start and some bookmakers on Wednesday night made United 4/1 to win the game.