The public callout also got the attention of Carlyle, one of the world’s biggest private equity firms, which moved quickly to encourage a deal between the two sides and urged Mr. Braun to reach out to Ms. Swift, according to four people close to the situation.

Carlyle’s intervention — which people in both camps say has brought the bitter fight closer to a resolution — says as much about the zeitgeist as it does about private equity. At a time of public outrage over corporate greed and a heightened awareness of gender-based power dynamics, the 29-year-old Ms. Swift was able to turn a commercial dispute into a cause célèbre.

According to the four people close to the discussions, a deal could take various forms, including a partnership or joint-venture arrangement. But the artist’s ideal outcome — and probably the only one she will accept, according to her team — appears to be a sale that would give Ms. Swift possession of her master recordings from Big Machine. Such an agreement could well cost her hundreds of millions of dollars.