If Facebook Inc.’s new digital currency goes according to plan, it could one day compete with payment giants Visa Inc., Mastercard Inc. and PayPal Holdings Inc. But for now, all three are set to work with the social-media company on the venture.

The currency, called Libra, will launch as soon as next year. It’s what’s known as a stablecoin, one that can avoid massive fluctuations in value so it can be used for everyday transactions. Industry experts and insiders say the payments companies want a seat at the table to help shape the new currency.

“It’s not unusual for the...