The reports said investigators for the newspaper allegedly hacked into Brown's personal bank account and accountant's records, and his lawyers were duped into giving details from his file to a conman working for the newspaper.
The Sunday Times is owned by News Corp., founded by Rupert Murdoch.
News Corp. also owns the News of the World, which has ceased publication because of a phone-hacking scandal.
The Sun, another newspaper owned by the Murdoch organization, is suspected of using confidential medical records when it reported in 2006 that the Browns' baby son was diagnosed with cystic fibrosis, both reports said.
Many other Labor Party leaders, including former Prime Minister Tony Blair, were also targeted by illegal News Corp. spying, The Guardian reported.
Current Prime Minister David Cameron was warned about the consequences of hiring the News of the World's ex-editor, the country's energy secretary said.
"I certainly raised it with Nick and Nick raised it with the prime minister," Energy Secretary Chris Huhne said. "It was very clear from what I said that big reputational risks were being run. Either Andy Coulson was complicit in criminal acts or he was an incompetent editor."
Cameron made it clear the job was "a personal appointment of the prime minister," the report said.
Rebekah Brooks, who edited the newspaper from 2000 to 2003, has contacted Scotland Yard and volunteered to discuss the scandal, it said.
Coulson and Clive Goodman, the newspaper's former royals editor jailed in 2007 after pleading guilty to phone hacking, were arrested Friday in connection with an investigation into payments the newspaper allegedly made to police.
Both were released on bail pending a court hearing in October.