Dean Henderson has had a run in the side in the absence of United’s number one, who has been inconsistent this season.
He impressed in Sunday’s 2-0 win against Premier League leaders Manchester City but was criticised for allowing an injury-time header to beat him as AC Milan snatched a 1-1 draw in the first leg of the last-16 tie in the Europa League on Thursday.
Asked on Friday about De Gea’s availability for the visit of West Ham, Solskjaer said: “David is back in Manchester. He is quarantining or self-isolating.
“He’s just touch and go for Sunday, I think. I can’t really know when he came back, so I’m not really sure if it was Monday or Tuesday that he arrived back in Manchester.”
Current regulations allow travellers from Spain to be released from quarantine if they test negative for Covid-19 after five days of isolation.
Pushed to clarify De Gea’s situation, the United boss added: “I think the rules are that he’s got to stay at home.
“He’s not training with us anyway. It’s different if you travel away and compete so, no, he’s not been training since he came back.”
United, second in the Premier League, are counting the cost of a spate of injuries in forward positions.
Anthony Martial is a doubt after a hip complaint forced him off at half-time on Thursday, compounding the absence of fellow frontmen Marcus Rashford and Edinson Cavani.
“Anthony, he got a whack on his hip so that’s another forward that we need to scan and have to look at,” Solskjaer said.
“Definitely hope Marcus is back for next Thursday in Milan. I’m not sure if he’ll make Sunday, but fingers crossed because we will need him.
“Edinson I don’t think will be ready for Sunday.”
Paul Pogba, Donny van de Beek and Juan Mata are also set to be unavailable to face the high-flying Hammers.