After making a high-profile move from Middlesbrough to United in 1989, Pallister was still kicking his heels at a local hotel with fellow new-boy Paul Ince when he was invited to watch the Red Devils' youth team in action.
Sir Alex Ferguson was also present and prior to kick-off advised the duo to pay particular attention to a youngster on the wing who went by the name of Ryan Wilson.
"The gaffer said he was going to be a star," said Pallister.
"I thought, 'That's a big statement, he's only a 15-year-old kid'.
"But there he was, this spindly-looking lad running amok up and down the left-hand side playing against opponents who were 16 or 17.
"He was absolutely terrific."
Within two years, Pallister and the 'kid', who had changed his name to Giggs after his parents split, were team-mates.
That season they won the League Cup, the following year the Premier League, one of four they were to be involved in, two of which turned into Double-winning campaigns.
Now retired for 12 years and carving out a niche for himself as a TV pundit, Pallister can only watch on with scarcely believable admiration as the 'teenager' prepares to turn 40.
He may no longer be quite able to be compared with the 'piece of silver paper floating on the wind' as Ferguson once did, but, mostly in central midfield, Giggs is still able to produce high-class performances that make him worthy of a place in David Moyes' squad.
"It's about time he retired, isn't it?" laughed Pallister.
"He is keeping young kids out of the game.
"What can you say about him? I still love watching him play.
"I obviously played with him for seven years, watched him mature and become the player he is now - one of the best to ever don the shirt of Manchester United."
Giggs' records are the stuff of legend.
Thirteen Premier League titles, two European Cups amid four final appearances, four FA Cups, three League Cups. And a staggering number of games for his one and only professional club.
One of Moyes' first tasks after replacing Ferguson as manager was to offer Giggs a first-team coaching role.
Many believe it will be the initial step towards Giggs one day taking over as manager in his own right.
For now, he is combining roles. Observing from the bench on occasion, sometimes in the stand. But also on the pitch, where he has done so much damage to opposition teams down the years.
And from what Pallister has seen, he is still capable of causing it.
"He is still making an impact," said the former England star.
"He can still play on that left-hand side of midfield, never mind centre-midfield, and still make a difference.
"We all recognise Ryan is not going to play as many games as most players in that squad, but when he does play, he is fit and ready.
"David Moyes has already used him. And believe me, there is no sympathy vote from him.
"Ryan is getting used because he can play an important part in this Manchester United team."