PayPal Holdings Inc. is nudging its customers closer to mainstream banking services.

The San Jose, Calif.-based payments company has been reaching out to groups of customers in recent months with an offer to add basic banking features to their PayPal digital wallet. The features include Federal Deposit Insurance Corp. insurance for balances up to government-set limits, a debit card that can be used to withdraw cash at ATMs and the ability to add funds to accounts by taking a photo of a paper check or by having employers direct-deposit...