The number of on-line banking users in the United States will more than double, to 2.1 million by 2000, according to a report from Meridien Research.

The report, based on responses from about 100 of the largest banks and financial services companies in Japan, Europe, and the United States, also predicted that the number of consumers using on-line financial services worldwide would quadruple, to 40 million by 2000.

In general, U.S. banks, brokerages, and insurance companies trail European providers in signing up on-line users.

For instance, 19.5% of French households and about 9% of Swiss households use PCs for banking or other financial services. By contrast, 1.9% of U.S. households do either banking or brokerage on PCs-about half are banking-only users.

By 2000 the United States should be closing the gap; about five million consumers are expected to be managing finances on-line by then.

France's growth rate "has slowed a tremendous amount," said Octavio Marenzi, research director in the electronic delivery services division of Needham, Mass.-based Meridien. "The U.S. is catching up pretty quickly."

Mr. Marenzi said the survey showed Internet-based financial delivery is not as common as many would believe.

"Although the World Wide Web has received tremendous attention as a service delivery channel, the Internet accounts for only slightly over 10% of on-line users worldwide," he said.

But more institutions will be providing Internet-based services over time, as direct dial-in and commercial on-line networks such as America Online increasingly are tied to the sprawling computer network.

Mr. Marenzi added that smart cards are becoming common tools for home banking. In Germany, about 25 million cards issued by banks are slated to be used for identification and authentication in home banking. Smart card readers are in about 500,000 French households.

Other researchers, such as Jupiter Communications and Forrester Research, have said the U.S. on-line banking market is already at or near two million users, on its way to as many as 10 million in 2000 or 2001.

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