Cousins' agent, Jeff Schwartz, told ESPN.com's Adrian Wojnarowski the deal is for more than the veteran minimum at $3.5 million.
Point guard Rajon Rondo signed a veteran minimum's contract shortly after Cousins, according to Wojnarowski and Ramona Shelburne. His deal is for two years and $2.6 million, with a player option for the second year.
Cousins signed a one-year pact with the Golden State Warriors last summer. He was coming off a torn Achilles suffered in January 2018, when he played for the Pelicans.
After missing the first three months of last season while rehabbing, Cousins made his debut in January and appeared in 30 games for the Warriors. The four-time All-Star struggled to adjust to his new surroundings and to find his form post-injury.
Cousins averaged 16.3 points and 8.2 rebounds per game, but his 27.4 three-point percentage was his worst since 2014-15. He also struggled on defense when opposing teams tried to take advantage of his conditioning and diminished speed.
Despite those struggles, Cousins seemed likely to take comfort in appearing in the playoffs for the first time in his nine-year career. But during Game 2 of Golden State's first-round series against the Los Angeles Clippers, he suffered a torn left quad that kept him out for 14 games.
Cousins returned for the NBA Finals against the Toronto Raptors. He was inconsistent in six games but did have flashes of his All-Star form. The Alabama native had 14 points and six rebounds in Game 5 after he took over when Kevin Durant injured his Achilles.
That marked two straight years in which Cousins suffered a serious injury to his left leg. It's fair to wonder which version of the center the Lakers will get next season.
If Cousins is the player he was in 2018-19, he can still be an effective asset in the right system. If he can revert to his pre-Achilles injury form, the Lakers will have signed one of the best big men in the NBA.
One year after being linked to the franchise, Cousins will partner up with LeBron James and Anthony Davis.
Cousins and Davis already have a rapport from their time together with the Pelicans. Cousins' arrival in Los Angeles also allows Davis to play his preferred spot at power forward, though he can still moonlight at center as head coach Frank Vogel needs.
Cousins is a high-risk, high-reward investment for general manager Rob Pelinka, but he's one the team can afford to take as it tries to get back to the postseason for the first time since 2012-13.