De Gea pulled up with a muscle problem during Spain's 1-1 draw away to Sweden in qualifying for Euro 2020 on Tuesday and is expected to be sidelined for a few weeks.
"David needs a scan,” Solskjaer told Sky Sports. "I think he'll be out."
Pogba was absent for France during the international break due to the recurrence of an ankle problem that has seen him start just once in the Premier League since August.
The World Cup winner played 90 minutes against Rochdale in the League Cup and Arsenal last month, but missed United's latest defeat at Newcastle just before the international break.
"He came back and played a couple of games, maybe played through the pain barrier," added Solskjaer.
"He had a scan after the Arsenal game and maybe needed a few weeks' rest in a boot, so hopefully he won't be too long, but he won't make this game, no.”
Solskjaer is, though, hopeful that Aaron Wan-Bissaka and Anthony Martial will make their return from injury against Liverpool, even if they are not yet fit enough to play 90 minutes.
"Hopefully Aaron and Anthony will last through the training this week and be available for selection. If it's for half a game or 30 minutes, I don't know, but let's see where they're at," said Solskjaer.