Federal investigators were closing in on Greg Lindberg. FBI agents confronted the North Carolina insurance tycoon last year as they probed whether he tried to bribe a state regulator. In March, officials obtained a sealed warrant for his arrest. His attorneys were negotiating his surrender.

Mr. Lindberg also had something else on his mind—the comings and goings of a number of women he was dating, interested in dating or, in at least one case, cultivating as an egg donor for his future offspring.

Mr....