The 14-year-old was watched by his father at United's Carrington training ground on Thursday, nine months after he had a trial at Chelsea.
Brooklyn's younger brother, Romeo, 12, has previously trained at Arsenal's academy.
A one-year-old Brooklyn in his father's arms back in 2000
A source told the Manchester Evening News: "He was there with his dad and took part in a practice session with one of the academy sides.
"It was a chance for him to impress and if he's done well then no doubt the club will look at taking him on."
Born the day after his father assisted Dwight Yorke's goals in United's 1999 2-0 quarter-final Champions League win over Internazionale, Brooklyn has keenly followed football.
He was first introduced to the Old Trafford crowd in 2000 during the champions' lap of honour, when he was starstruck by United mascot Fred the Red.
A year later, he scored his first goal at the Theatre of Dreams at the tender age of two.
Brooklyn is the same age as dad David was when he joined United as a schoolboy in 1989.
In January, he could be seen celebrating Tom Cleverley's opener for United at West Ham in the FA Cup third round, only he and his father left early and missed Robin van Persie's equaliser.