The turnaround is remarkable, given United finished last season 33 points adrift of Liverpool, as the Anfield club ended their 30-year wait for the title.
Asked if United should be ahead of Liverpool on a regular basis, Solskjaer said: "Of course. We know what they've done the last three or four seasons.
"They've been top of the league, they've won the Champions League and they've had some amazing seasons.
"That's the challenge for us, to have that consistency now that we've moved up the table and that we also challenge Manchester City."
But Solskjaer said he could not care if Jurgen Klopp's side managed to sneak into the Champions League next season or not.
"We don't go out there thinking about that," said Solskjaer. “We're just thinking about winning ourselves.
"Whatever that does to Liverpool cannot be of our concern. Throughout the season you have games you look forward to.
"This Liverpool game at home is one of the biggest games of the season.
“It doesn't matter if we're first or second or third or fourth. No matter what, this is a massive, massive game."