The Times reports today that the government is in favour of a sale of the bank, which was nationalised to save it from collapse.
The paper says that prime minister Gordon Brown is keen to sell Northern Rock back into private ownership at a profit before an election and avoid the Conservatives taking credit for any such deal.
Tesco already offers consumers a range of Tesco-branded financial products, including savings accounts, credit cards and loans.
Opinion at the moment is that consumers see supermarkets such as Tesco and Marks & Spencer as more trustworthy than banks, and these brands are keen to capitalise on the sentiment.
Prior to the bank being nationalised, Virgin had tried to buy Northern Rock. The Times reports that it is also considering making another bid, along with private equity firms.
There are plans to split Northern Rock in two, in a deal that would make its sale more attractive.
This would leave "toxic" loans in the hands of the public, with one half of the business responsible for paying back money lent it by taxpayers.
The plan must be approved by the European Union.