ESPN sources say Nash was swayed to join the Lakers due to a push from Kobe Bryant.
The move to Los Angeles will allow Nash, who spent the past eight seasons with the Suns, to stay close to his three children in Phoenix.
According to sources, Nash will receive a three-year deal for $27 million. In return, the Suns get four draft picks - first rounders in 2013 and 2015 and second rounders in 2013 and 2014.
Steve Nash's agent Bill Duffy said the deal was finalized Wednesday about 6 p.m., PDT.
Nash has never made it to the NBA Finals. He was last in the Western Conference finals against the Lakers in 2010, when the Suns lost in six games and Los Angeles went on to win the NBA title.
The Associated Press contributed to this report.