Newcastle United are targeting the defender this season, though there's claims Liverpool won't allow him to leave.
However, Brown told Football Insider: "His chances are going to be limited at Liverpool.
"I picked Quansah out when he was in the youth team and I always thought he had the makings of becoming a top player.
“He’s got to a certain level in his career, but now I think he needs to push on again. To do that, he’s got to be playing football at the top level.
"With the options they’ve got at Liverpool at the moment, it doesn’t look like he’ll be a regular any time soon, especially while they’re looking to sign new players there.
"So if he’s got the opportunity to go and play for Newcastle, I hear he wants to do that. He knows what’s best for his career, and at this stage is being able to play every week.
"It would be a tremendous move for him and for Newcastle, they’d be picking up a very promising young player if they could get it over the line."