Specifically, they wonder, why should anyone use their branches when customers can move money around, check their balances or pay bills without leaving their desks.
It has already happened in a big way with phone banking. Computer-based banking, which has the added advantage of displaying balances and statements on screen, must surely make a further dent in traditional bank activities.
In the US, according to the Online Banking Report, almost 5m people have used Internet banking facilities, while 22m are expected to do so by 2001. In the UK we are well behind.
Prestel On-line, the Internet Service Provider, surveyed 100 of its subscribers and found that 86 per cent have yet to use banking facilities on the Internet. But 94 per cent said that they expect to have done so by the year 2001.
The banks, not wanting to be left behind, and have been moving into computerised banking. The first was the TSB, which set up on CompuServe, and in the last two years several other banks have launched "PC banking" services.
These are not Internet-based - customers need special software and have to dial a dedicated number - but these banks do pave the way for the switch.
In January last year, the Royal Bank of Scotland became the first UK bank to offer Web banking, and there has been a gradual build-up in activity since. Nationwide, the UK's largest remaining building society, has one of the more advanced systems, which allows customers to view transaction histories, transfer money between accounts, and set up bill payment by direct debit. It says 50,000 UK customers have used the service.
The Co-op and Barclays have also moved to the Web, and Citibank launches its service today. Still to come are Lloyds TSB and NatWest, which are hoping go live later this year, and Midland, which starts its trial in the autumn.
Some banks, including Barclays, are keeping a PC service alongside the Web service. The functionality is slightly greater, though the real advantage is psychological; some customers just do not trust the Internet, even hefty security systems are now being offered.
For most customers the Web will prove the most convenient option, because it can be accessed whenever they are online. For the banks, the online channel represents a major opportunity to cut costs. Internet transactions are four times cheaper than those conducted through physical branches.
The Internet also presents a cheap method of expanding into new markets. Nationwide is targeting continental Europe (without physical branches) once the Euro currency has been established, while Citibank's expansion in the UK is almost entirely "virtual".
Citibank is claiming to offer saving interest rates that are on average 11 per cent higher than those offered by the high street banks, as well as allowing customers to transfer money between Citibank accounts in different countries.
"Global retail banking" is constrained by national regulations, but it will spread fast if these are lifted. The next stage is the "pure Internet" bank. In the US, Atlanta Internet Bank (www.atlantabank.com) offers much higher rates to savers because it has no physical infrastructure to support.
Ben Schiller works for Net Profit newsletter.