The undercover reporter from The Sun paid 拢3,000 for data including credit card numbers and supposedly secret passwords, which were gained from call centre workers.
The revelations are likely to amplify public disquiet about the outsoucing of call centres to cheaper Indian operations and the handling of data by their banks.
The Sun reporter bought the records in Delhi from an Indian man who said that he obtained them from a number of different contacts working in call centres.
The Sun's report said: "Targets have accounts with leading High Street banks and building societies including HSBC, Barclays, Halifax, Lloyds TSB, NatWest, Abbey, Woolwich, Royal Bank of Scotland and Nationwide."
The man used to train call centre staff in India and told The Sun reporter, Oliver Harvey, that more details were available and that he could supply 200,000 identities a month.
However, Halifax, RBS, Nationwide and NatWest have since told the BBC that they did not have call centres in India so were unlikely to be affected.
Barclays and HSBC have launched investigations and said they would be assessing the level of risk to customers.
"We are taking the allegations very seriously and are investigating the matter with the aim of safeguarding our customers' details," a spokeswoman for Barclays said.
The Sun has handed over its evidence to the City of London Police, which has said it is investigating the allegations.
Oliver Shaw, detective chief inspector at City of London Police, said: "While the allegations are very serious, we would like to remind readers that instances of this kind are still relatively rare.
"Our advice is never to respond to unsolicited telephone calls regarding your account security details. When internet banking, always make sure that you have logged on to a genuine financial site and never click on pop-up boxes."
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