But there is no way of paying bills direct from either of the accounts. A current account will follow later this year but until then customers will have to use their accompanying cashcards to draw out money, or use either of the two Sainsbury's credit cards on offer.
Customers who want to borrow money can use the credit card but this is a relatively expensive option. Personal loans, not yet available from Sainsbury's, would be a cheaper way to borrow.
But Sainsbury's is not alone. Other supermarket bank accounts are showing similar signs of an unseemly rush to get up and running. Tesco's revamped Club- card Plus account pays 5.5 per cent interest on cash balances and customers can use the card to pay for shopping right away. But Clubcard Plus is not a free-standing account; it has to be fed by regular transfers from another account elsewhere and it cannot be used to pay bills other than at Tesco. Customers can use the Clubcard to draw money from NatWest cashpoints, and they can borrow an amount equivalent to the sum they feed into the account each month, but there is no credit card.
Safeway's ABC Bonus Account is a self-contained bank account. It pays 5 per cent interest on balances up to pounds 600. Customers receive a Visa Electron instant debit card, which can be used to pay bills from their accounts at Safeway stores and in any of the 70,000 shops countrywide that accept the Electron card. This doubles as a cash dispenser card, but again, it is not a credit card. In fact, none of the supermarkets can yet compete with established banks and building societies.
But they will appeal to customers attracted by the rates of interest. Some shoppers will also change their shopping habits to get the account that suits them best. So Asda may live to regret its determination not to get drawn into the field.
But the real test of supermarket banking will be whether the stores can tap their captive market, make use of the detailed database revealed by customer shopping bills, and build on their brand image to cross-sell other financial products such as personal loans, pension plans, mortgages and insurance policies.
On its own, a basic money- transfer service is neither profitable nor popular, as the clearing banks have found to their cost during the past two decades. And, revealingly, neither Marks & Spencer nor Virgin Direct has been tempted to offer a banking service. These two companies are as different as chalk and cheese, the one apparently reclusive, the other extrovert to a fault. But both moved into the personal finance market at the right time when many savers were increasingly dissatisfied with the services offered by established banks and insurance companies.
Now the financial services market is recovering steadily from the confidence- sapping fiasco of mis-sold pensions, recession, falling property prices and concerns about whether endowment policies would be able to pay off mortgages. So the supermarkets are spotting the opportunity to sell branded financial services.
But will they be quite so enthusiastic when the economy slows, share prices fall, and customers start to feel unwilling or unable to keep up their payments? In the meantime, the shops will also risk alienating ordinary customers if they try too hard to cherry-pick.