The potential transaction was for a stake of "up to 15 percent" of the three-times winners of the Rugby World Cup, the source said, asking not to be identified because the negotiations are private.
The talks with Silver Lake were first reported by Sky News, which said the deal for a 15 percent stake would value the team at about $2 billion. Sky said a deal had not been concluded, but could be announced as soon as this month.
An investment in one of the most famous brands in world sport would expand the buyout giant's $75 billion portfolio of companies, most of which are technology and sports venues and teams, according to its website.
Representatives from Silver Lake declined to comment.
Charlotte McLauchlan, director of communications for New Zealand Rugby, told Reuters the organisation was "on a path to look at what an investor partner for New Zealand Rugby might look like - which is a very exciting prospect for us".
She declined to comment any further.
Last year New Zealand Rugby launched a review into how the game in the rugby-mad country is run and financed, saying it could include private equity investments.
In 2019, the California-based firm acquired a 10 percent stake in City Football Group (CFG), which owns reigning English Premier League champions Manchester City and teams in the United States, Australia and China.