PayPal's split from eBay was supposed to sweeten its appeal to merchants, but it’s also stoking rivals in the payments industry — in particular Visa CEO Charlie Scharf.

"I've been very consistent with what I have said about PayPal," Scharf said Wednesday during the Bernstein Strategic Decisions Conference in New York. "People talk about 'frenemies,' and there is no such thing in my mind. You are either one or the other."

Scharf took the stage several hours after PayPal's president and CEO, Dan Schulman, attempted to set a much more conciliatory tone as his company seeks to ally with its longtime competitors.

Scharf, who sent a similar message last week at a conference hosted by JPMorgan Chase, said he respects PayPal executives and what that company has accomplished, but slammed PayPal's longstanding practice of urging consumers to use Automated Clearing House transfers instead of payment cards to fund their PayPal accounts. "What they do as a business is not good for us or for our clients," he said Wednesday.

Even though half of PayPal's transactions move through the card networks, the alternative payments company is relentless in its push to get consumers using the cheaper ACH method, Scharf noted for the second time in recent weeks.

"That trumps everything, and it is not something we like or support," Scharf said. "We have sat idly by for a long time, but the world is different today and there are many options available in the digital world."

These options include mobile wallets such as Apple Pay, which lets consumers link payment cards directly, as well as the card network's own Visa Checkout online payment system.

Visa would "like to have a different relationship" with PayPal, Scharf said. "We will either have that relationship, or we will be fierce competitors, which we haven't been."

Scharf did not elaborate on when that "fierce" attack on PayPal would come, but he said Visa's strategy for such a move is already in place.

For his part, Schulman is spending a fair amount of time these days reminding the card networks that PayPal has delivered plenty of business to their rails, and wants to build on that foundation to become a true ally.

"We drove a tremendous amount of traffic to the issuing banks and networks around the world with 14 million merchants, most of them small and midsize," Schulman said. "Without us, they wouldn't be utilizing a credit card or being online and driving a huge amount of incremental traffic."

Part of the conflict is that both companies' expansion strategies have set them on a collision course. Visa is placing its investing in becoming more prolific in mobile and digital payments, while PayPal is expanding its platform to look and feel like a full-service network.

"We now have a very expansive strategy and vision," Schulman said. "We are clearly moving to be more than a button on a website, and we have a strategy to be part of the everyday consumer's financial life, not just an occasional transaction in e-commerce."

PayPal wants to be the underlying operating system and platform powering digital commerce going forward, Schulman said. "We want to be 100% share of checkout, and we want to provide a platform for retailers to create value proposition and we will power that for them."

PayPal used to be a company that "wanted to do everything alone and did not want to partner," Schulman said. "But I believe fundamentally that this is not a zero sum game and we are at the very beginning of an addressable market and the ability to use our platform to partner with financial institutions and tech companies is fundamental for us to achieve our vision."

Visa's digital efforts have had to share the spotlight with other major projects such as the U.S. conversion to EMV chip cards and the reunification with Visa Europe. However, Visa also made the important decision this year to open its network to third-party software developers, putting it in direct competition with PayPal's Braintree operation.

Schulman acknowledged the lingering friction PayPal has over its longtime practice of encouraging ACH over card payments, but he said he has been visiting with financial institution and network executives to provide a better understanding of how PayPal is evolving. "We are showing them exactly what we are planning to do."

Schulman admits that some past reservations these companies had about working with PayPal stemmed from his company's practice of not sharing transaction data as openly as it could have.

"I am not even sure why that happened," Schulman said. "There is no good reason to not share data with the issuers so that it can be on the credit card statement so the consumer knows which retailer is involved, and not just PayPal."

The areas that issuers and retailers perceive as friction with PayPal "will go away" in the future and recent discussions along those lines have been positive, Schulman said.

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