Brown, who spent five years coaching the LeBron James-led Cleveland Cavaliers, compiled a 272-138 record with playoff appearances each year there.
He took the Cavs to the NBA Finals in 2007, where they were swept by San Antonio.
Brown has some big shoes to fill in replacing Jackson, 65, a Hall of Famer with 11 total NBA championships and the only coach in league history to lead a team to three consecutive championships three different times.
"In terms of replacing Phil, I'm not him. I'm not going to be him," Brown said. "But I do have mentors that I reached out to to talk about this."
Brown was named NBA Coach of the Year after the Cavs went 66-16 in 2008-09. Cleveland, however, lost in the Eastern Conference finals to Orlando, and he was fired following the next season after the Cavs exited the playoffs in the second round.