ESPN.com's Pierre LeBrun is reporting the deal is worth $58 million.
Under the NHL's collective bargaining agreement, the new deal cannot be signed until July 1.
The 26-year-old Quick won the Conn Smythe Trophy after the Kings' Stanley Cup-winning playoff run and also was a Vezina Trophy finalist and an NHL All-Star this past season.
Quick, who had one season left on his contract, compiled 16 wins in 20 postseason games while leading all playoff goaltenders with a 1.41 goals-against average (third-best all-time for goalies with at least 10 games played) and a .946 save percentage (tied for third-best all-time for goalies with at least 10 games played). He also tied for first with three postseason shutouts.