United have already lost to Liverpool this season and a second defeat would add more pressure on Moyes.
Pallister, who lifted four Premier League titles during his nine years at Old Trafford, told the Daily Star: "It's a massive match. There's a lot of pride at stake.
"They are old nemeses - and they'll both be reeling a little bit after weekend defeats.
"But there's too much experience in the United dressing room to let the City loss worry them unduly.
"It's a long season and they're not too far behind in the Premier League, even with the start they've had.
"I don't think there's any reason for panic just yet. If they'd been playing a lot worse I'd be more concerned but the signs are there.
"The players will be hurting - they know they didn't play to their standards.
"The manager will be demanding more and the fans will be demanding more - but the players will be demanding it of themselves to produce something against Liverpool."