Williams, a three-time All Star, met with representatives from both teams this week and ultimately chose the Nets franchise that has moved from New Jersey to Brooklyn and will play in a new arena in the coming season.
"Made a very tough decision today," Williams tweeted, before posting a link to the Brooklyn Nets logo.
Williams would sign a five-year deal worth $98 million, local media have reported.
Having played parts of the last two seasons with the Nets after being traded from Utah, Williams is set to join a team on an aggressive recruiting drive in the hope of making the playoffs for the first time since 2007.
The Nets have been linked to a trade with Atlanta for All Star Joe Johnson, and may also land disgruntled Orlando center Dwight Howard, who requested to be traded to Brooklyn.
Williams played in four postseasons with Utah and averaged 21 points and 8.7 assists for the Nets last season.