The changes will amount to nothing less than a revolution, in which some of today's leaders will become laggards while some of the back markers will come to the fore.
Nationwide, now Britain's biggest building society, has launched a Web site for FlexAccount customers, providing account balances, automated bill payments and similar services. First Direct is currently operating on a trial basis, but expects to roll out the full service - available to all 650,000 customers - in the autumn.
Institutions have been holding back the introduction of Internet banking until now because of fears over security, but as these have now largely been overcome we can expect to see increasing numbers of on-line banking operations making their debut.
The attraction for the customer may be the convenience of being able to check a statement on a Sunday morning or calculate the cost of a mortgage before a weekend house hunting.
For the banks and building societies themselves the driving force is not customer convenience but cost reduction, according to Bob Scott, director of electronic commerce at Cap Gemini, Europe's largest computer services company.
"Just look at the figures," he says. "The latest research shows that for every 100p it costs to carry out a transaction over the counter it costs 80p to do it in a manned call centre where you talk to someone on the other end of the line, and 60p for an automated service using a touch- tone phone and voice prompts.
"What's the comparable figure for providing that transaction using an Internet-based banking system? About 5p. Even if that calculation turns out to be on the low side, and the true figure is 8p or 10p, the savings are of such an order of magnitude that they cannot be ignored."
The snag is that even a small increase in electronic business could, for example, push large numbers of bank or building society branches, already operating at narrow margins, from profit into loss. So the reduction in branches already experienced by the big banks could accelerate.
"If you open an account with your local branch, but then do all your transactions on the Internet, who makes the profit?" asks Bob Scott. "Is it the physical branch which you visit once in a blue moon or the virtual one - operating between your PC and the bank's data centre - that you deal with all the time? It all depends how the bean counters at head office count the beans"
On a more positive note, some of the smaller regional societies which could not afford to build a network of physical branches will be able to gain a national presence and win market share - for what are often highly competitive products - from their electronic distribution channels.
And although most core Internet technology comes from the United States, many of the systems being adopted in the UK and elsewhere in Europe are being developed on this side of the Atlantic.
Cap Gemini is involved in many of them, including the OpenBank from Banco Santander which has been running as a large scale pilot for about two years - giving both the bank and the developers a treasure trove of information about how the system works - and, equally importantly, how customers use it.
First Direct's new on-line service is based on systems developed by ICL which have been highly praised by banking technology specialists and which will be taken up by many other institutions in the future.
A recent survey undertaken for ICL by MORI, the market researchers, suggests that 31 per cent of adults expect to use electronic banking within the next five years. But in case you're worried that you might miss out on the best deals because you aren't yet connected to the Net, don't worry.
At present, you think of the telephone, the television and the computer as three different gadgets, but technology is fast blurring the distinction.
Handheld computers with built-in cellular phones have been on sale for some time, and while the notion of being able to check your bank balance on your wristwatch is still science fiction, the time when it will be routine is much nearer than you think.
First Direct's on-line banking service uses an Internet technology called Java which is what the boffins call "device independent". Right now, you use it on a computer, but it is already being transplanted onto mobile phones and hand-held personal organisers. When interactive digital television arrives, Java will operate through the set-top box, bringing armchair banking to millionsnReuse content