At prime minister's questions last Wednesday, Mr Brown said Tory plans for inheritance tax cuts would help millionaires but cost public service investment £2bn.
He told Mr Cameron in the Commons: "I have to say, that with you and Mr Goldsmith your inheritance tax policy seems to have been dreamed up on the playing fields of Eton."
Mr Brown was referring to Zac Goldsmith, the millionaire environmentalist and prospective Tory MP who has admitted having non-domicile tax status.
Other Conservatives to attend the exclusive school in Berkshire include Mayor of London Boris Johnson and the shadow leader of the Commons Sir George Young.
'Lurch to the left'
Mr Cameron said on Sunday there were many members of his shadow cabinet who attended state schools, including shadow foreign secretary William Hague, shadow defence secretary Liam Fox and party chairman Eric Pickles.
He said: "Frankly I think the country is more interested in who are these people, are they any good, have they got the right ideas, will they take the country forward, have they got the energy and vigour and dynamism that we so badly need?
I don't care whether David Cameron went to Eton or not - what I do care about is the fact that I think consistently he is making the wrong calls
Alistair Darling Chancellor of the Exchequer
"And the answer to those questions I believe is yes. But the best way to test it, you know what? Have an election."
However, Mr Darling told the BBC's Andrew Marr show that he judged people "not on what school they went to but on the choices they make and the priorities they face".
The Chancellor added: "My view is that to hand over a large sum of money to the 2% of the largest estates doesn't make any sense.
"I don't care whether David Cameron went to Eton or not - what I do care about is the fact that I think consistently he is making the wrong calls, the wrong judgements, and represents a serious risk to the country."
Shadow chancellor George Osborne told the programme that his party's inheritance tax plan would be introduced over time "when we can afford it".
He added: "Frankly this is a Labour Party political attack, and if the Labour Party wants to say to people, 'Don't aspire to get on in life, don't aspire to own your own home, don't aspire to leave anything to your children' then so be it - it is part of their lurch to the left."
Asked by Sky News about the row, Liberal Democrat leader Nick Clegg, who went to Westminster public school, said: "I don't care where someone went to school. I don't find that relevant.
"What I do find relevant is whether politicians have the right ideas to restore fairness to Britain."
The row came as two new polls suggest that the Conservatives would secure a majority at the next general election.
A YouGov survey for the Sunday Times has the party on 40% - 13 points ahead of Labour on 27%.
An ICM poll for the Sunday Telegraph also has the Tories on 40%, but 11 points clear of Labour on 29%.
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