Pogba has missed five of United's last seven games because of a foot problem and, despite a week of warm weather training in Dubai, will not be available to face Jurgen Klopp's side at Old Trafford on Sunday.
United are awaiting results of a scan on David de Gea's abductor problem but Solskjaer is not convinced the goalkeeper will recover in time to play.
"Paul had an injury, he came back and he worked really hard," Solskjaer told Sky Sports.
"He came back and played a couple of games, maybe played through the pain barrier. 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.
"David needs a scan. I think he'll be out. It certainly looked like it anyway so it's just one of those things."
Jesse Lingard has also been ruled out with a hamstring injury but there is hope Aaron Wan-Bissaka and Anthony Martial may be able to play some part against Liverpool, who are eight points clear at the top of the Premier League after eight wins from their first eight games.
"Hopefully Aaron and Anthony will last the training this week and be available for selection," said Solskjaer.
"If it's for half a game or 30 minutes I don't know but let's see where they're at."