The 14-year-old, who has spent spells with Chelsea, Arsenal and QPR, has been afforded an open invitation to return as the club looks to run the rule over David's son.
The 14-year-old has had previous stints with Chelsea and QPR, while he also trained with the LA Galaxy academy during his father's spell in the MLS.
The youngster took part in training sessions with members of the Manchester United youth set up at the club's Carrington training facilities this week.
Goal understands Brooklyn also had a recent trial with Arsenal and returned to Chelsea for a second stint before being offered an open invitation to return to United in the future.
Beckham was part of United's FA Youth Cup-winning side of 1992 before going on to amass over 400 caps for the club, winning six Premier League titles, two FA Cups and the Champions League.
Beckham's youngest son, Romeo, has also previously trained with Arsenal.