The private equity industry has long been dominated by the Wall Street elite: firms like the Blackstone Group, Kohlberg Kravis Roberts and Carlyle.

Now SoftBank, a Japanese conglomerate best known in the United States for its ownership of Sprint, is looking to muscle in on this lucrative business.

Under Masayoshi Son, the founder who still leads it, SoftBank has been a relentless deal maker for decades, shaking up industries and most recently unleashing a $100 billion investment fund focused on technology.

Despite Mr. Son’s reputation — he is sometimes compared to Warren E. Buffett — the Wall Street establishment was skeptical when SoftBank last year made its first big private-equity play, buying Fortress Investment, a midsize asset manager.