The deal is reportedly worth $18.25 million over four years, according to ESPN.
Jacks, the 11-time NBA champion coach, retired earlier this month after the Dallas Mavericks ousted the defending two-time champion Lakers from the second round of the playoffs.
Although Kobe Bryant endorsed Jackson assistant coach Brian Shaw for the vacancy, the Lakers replaced Jackson with Brown, a respected young tactician who has ample experience in big games -- even if he couldn't guide LeBron James to a title.
The 41-year-old Brown led the Cavaliers to the 2007 NBA finals and won 272 games in five years, becoming the most successful coach in franchise history. But the 2009 NBA coach of the year was fired last spring following the Cavs' dissension-filled exit from the second round of the postseason, and James left for Miami a few weeks later.
Brown spent the past season as an ESPN analyst. The Golden State Warriors also are considering him for their coaching vacancy.
Brown's background in defense apparently intrigues the Lakers, whose last two title runs were built on sturdy defense led by Bryant and 7-foot shot-blocker Andrew Bynum. Brown is a former assistant to Spurs coach Gregg Popovich.
Shaw, a former Lakers guard and a top assistant coach during Jackson's tenure, was the favorite candidate among the current Lakers, with Derek Fisher and Bynum joining Bryant in throwing their support behind their former teammate.
But the Buss family has a history of idiosyncratic coaching hires, often from outside the organization. When Jackson left the Lakers for a year in 2004, they replaced him with former Houston coach Rudy Tomjanovich, who resigned midway through his only season because of health issues.
Lakers general manager Mitch Kupchak has said the club is likely to return with largely the same veteran core that won the past two NBA titles before falling short this season. Los Angeles already has more than $85 million in salary committed to eight players for next season, likely meaning the Lakers will have the NBA's largest payroll again next season.
The Associated Press contributed to this report.