The Lakers' offer includes the second overall pick in Thursday's NBA draft and "might be the best offer the Kings can get" at this point, according to the person, although the Kings have publicly stated they are not interested in trading Cousins.
The Lakers would probably also have to give up Julius Randle to complete the deal, among other salary considerations. Randle sustained a season-ending broken leg on opening night last season after being drafted seventh overall.
Cousins, 24, averaged 24.1 points, 12.7 rebounds and 3.6 assists a game last season, his fifth in the NBA. He is one of the game's top centers and is an active defender, averaging 1.7 blocked shots and 1.5 steals last season.
Kings Vice President Vlade Divac was "very irritated" to read reports of the Kings' desire to trade Cousins, according to the Sacramento Bee.
Divac also denied that Sacramento Coach George Karl was pushing for a deal involving Cousins. A Yahoo Sports report said Karl and Cousins had a tenuous relationship at best. Karl became the Kings' head coach in February after Mike Malone was fired.
Cousins has three more years and $50.9 million remaining on his contract. He was the fifth overall pick in 2010.
Meanwhile, the Lakers continued to weigh their draft strategy, assuming they don't trade their No. 2 pick.
They haven't had a selection that high since taking James Worthy first overall in 1982, and are deciding mainly whether to take Duke center Jahlil Okafor or Ohio State point guard D'Angelo Russell.
Latvian center Kristaps Porzingis has less of a chance to be drafted by the Lakers but intrigued them with his workouts and unique blend of talent as a 7-foot-2 center with three-point touch.
The Lakers finished with the NBA's fourth-worst record last season (21-61) but jumped two spots in the draft thanks to a lucky lottery night last month.