Ivica Olic looks to be on his way back to Hamburg, with Andre Schurrle increasingly likely to join Wolfsburg.
The 28-year-old former Manchester City forward, son of Ian Wright and brother of Shaun Wright-Phillips, was released by Charlton at the end of last season.
"Since Bradley's potential was first recognised by Manchester City and the England Under-20s, he has proved to be a goalscorer of note," Red Bulls sporting director Andy Roxburgh told the franchise's official website.
"He'll add another dimension to our roster with his pace and technical ability."
Wright-Phillips also numbers Southampton and Plymouth among his former clubs and most recently played for Brentford, on loan from Charlton.