• Gordon Brown is questioned by Gillian Duffy
The PM's comments came as he was driven away from an event in Rochdale at which 65-year-old widow Gillian Duffy tackled him in front of the cameras about Britain's financial problems, taxes, student financing and immigration.
Unaware that his radio mike was still connected, Mr Brown told an aide that the encounter had been "a disaster" and said he should never have been made to speak with Mrs Duffy, adding: "She was just a bigoted woman."
Labour has changed its campaigning tactics in recent days to expose Mr Brown to ordinary voters more, after complaints that he had been appearing only in carefully controlled conditions in front of Labour supporters.
Immediately following her discussion with Mr Brown, Mrs Duffy said she would vote Labour and thought the PM was "nice".
However, in his car, a clearly angry PM was telling his aide: "That was a disaster – they should never have put me with that woman. Whose idea was that? It's just ridiculous..."
Asked what Mrs Duffy had said, he replied: "Everything, she was just a bigoted woman."
After hearing of the PM's comments, Mrs Duffy said he owed her an apology: "I'm very upset. He's an educated person. Why has he come out with words like that?
"He's supposed to be leading the country and he's calling an ordinary woman who's come up and asked questions that most people would ask him...
"It's going to be tax, tax, tax for another 20 years to get out of this national debt, and he's calling me a bigot."
• Gordon Brown says sorry after being caught out on microphone in an unguarded moment
Speaking later on BBC Radio 2's Jeremy Vine show, Mr Brown said: "I do apologise if I said anything which is hurtful and I will apologise to her personally."
As his comments were played back to him in the radio studio, Mr Brown put his head in his hand, and then said: "Of course I apologise if I've said anything that's been offensive and I would never put myself in a position where I would want to say anything like that about a woman I'd met.
"I blame myself for what is done, but you've got to remember that this was me being helpful to the broadcasters, with my microphone on, rushing into the car because I had to get to another appointment and they have chosen to play my private conversation. These things can happen, I apologise profusely to the lady concerned."
Shadow chancellor George Osborne told Sky News: "We have found out the Prime Minister's internal thoughts and I think they speak for
themselves and the Prime Minister has got a lot of explaining to do.
"The thing about general elections is that they reveal the truth about people...
"What people will see is the contrast between what he was saying publicly and what he was saying privately."
The comments came as the economy moved to the fore of the election debate, with all three parties admitting that the UK faces deeper spending cuts in the years after the May 6 General Election than have so far been detailed.
The independent Institute for Fiscal Studies yesterday accused politicians of keeping voters in the dark about the tax rises and spending cuts which will be needed to meet their targets for reducing the UK's record 163 billion national deficit.
The warning came as stock markets plunged due to the financial crisis in Greece, which has seen its sovereign debt slashed to junk status amid discussions of a eurozone bailout.
Mr Osborne said that Greece stood as a warning to Britain of what happens if a state fails to pay its debts. But Labour accused Tories of "economic illiteracy" for comparing the countries.
"Britain is not Greece. Greece is not Britain. We are very different economies," said Business Secretary Lord Mandelson.