The 19-year-old was the 55th choice of the draft in New York on Thursday after weeks of speculation about which team he would end up on.
Four-time NBA champion LeBron, 39, is probably entering the final year of his career and could become a free agent this summer by declining his $51.4m (£40.6m) player option when the window opens at the end of June.
However, that prospect seems remote given the four-time NBA champion said after the All-Star game in 2022 that his “last year will be played with my son”.
No father-son pairing has ever played together in the NBA before.
On Instagram, Bronny wrote “Beyond Blessed”, while his father posted a series of pictures of the pair and the caption “Legacy”.
Bronny has completed his first year of college basketball with the University of Southern California Trojans, where he averaged 4.8 points and 2.1 assists per game.
Unlike his father, who stands at 6ft 9in and is the NBA’s leading all-time scorer, teenager Bronny is 6ft 1in and is more of a defensive player.