The 18-year-old winger came to the fore during pre-season, but announced himself in the Premier League in style with a match-winning double at Sunderland on October 5 to leave manager David Moyes and the club's fans alike drooling.
However, Pallister insists the teenager has to be allowed to grow at his own pace despite the excitement surrounding him.
The former United defender, who was supporting the Manchester United Foundation Chevrolet Trax eBay auction, said: "You can't get too excited about him - and it's easy to.
"At Old Trafford and Manchester United, it's quite easy to see these players and all of a sudden, he's the new George Best or the new Ryan Giggs.
"At a young age like that, you don't want to heap too much pressure on him.
"What I will say is the kid has been fantastic. He has shown maturity beyond his years. He's making a real impact.
"He's fearless, he's not afraid to take the ball in tight situations, he's getting kicked and he gets up and goes back for more and he's got that bit of devilment in him as well, so he has got all the attributes.
"But to heap that kind of pressure on him saying he is going to be this and that, you wouldn't want to do that with the lad.
"You have got to let him develop in his own time. He isn't going to play week in, week out, I don't think, but he's making an impact and he has got everybody at Old Trafford really excited about the future and what he can bring to the team."
United are currently in the process of recovering from an ordinary start to the season by their standards with Moyes' defence a particular focus following three defeats in the club's opening 10 Barclays Premier League games.
Pallister believes the new manager is still searching for his best central defensive pairing with experienced duo Rio Ferdinand and Nemanja Vidic being pushed by Chris Smalling, Phil Jones and Jonny Evans.
However, he insists it is unrealistic to expect the same two men to play in every single game in the modern age.
He said: "You will have an idea which is your best partnership, but you can't expect them week in, week out to play every game. The demands are far too great now.
"Footballers today are more athletic than they were even in my time. These guys have six-packs, and the sports science that they work with these days, they are at the very peak of physical condition.
"Everybody needs a rest at times. The games come too thick and fast and at too much of a high intensity, especially when you play at United.
"Everybody wants to beat Manchester United, so these games get extra-tough when you play some of the lesser teams in the Premier League.
"I think David Moyes is working out who his best partnership is, but there are still parts to play for everybody."