Jack has signed a new contract that will expire in the summer of 2014.
The 19-year-old, who has been with the Dons since the age of eight, has made 46 first-team appearances and is a regular with the Scotland Under-21s.
Brown said: "His next move is hopefully not a moderate move to the Championship.
"If he moves to improve himself I'm hoping it will be to one of the top four or five in England. If he moves.
"But hopefully that will not be for a while yet, because we can use his services here. They are needed at the moment.
"A guy like that should be aiming for the top. He is a Scotland Under-21 international with a good number of caps and SPL appearances.
"He has the world at his feet and his next move should be Manchester United or Liverpool."
The Aberdeen manager, whose team host Dundee United on Saturday, said: "For a young lad he is very consistent, he doesn't have many off days and makes few mistakes.
"And he is very creative, he scored one of the best goals we have scored since Archie and I came in, at Inverness last season.
"I always considered him a midfield player but we are well provided in midfield and he has filled in well at full-back.
"His use of the ball is excellent and he has grit and determination on the tackle.
"He is quick and is two-footed. he is quite a rounded player for a young lad, it's hard to find fault."
Brown added: "He's a local boy, that's the ideal scenario, if you're team is a local team.
"We have a few local boys coming through so hopefully we can tie them down too."
Source: Team Talk