By Daniel J. McLaughlin
Privately-educated people make up such a small percentage of the population, but they dominate the top jobs.
A private education can cost thousands of pounds each year for parents who want to give their children an advantage in life.
John McDonnell argues that private schools are part of the inequalities in education that creates a "grotesquely unequal" society.
However, defenders of private schools say that they are "an enormous boon to this country".
McDonnell has backed a campaign by Labour Against Private Schools to effectively abolish private schools, the Independent reports.
The group is proposing the "Abolish Eton" motion at the party conference this weekend.
It would remove charitable status from private schools, impose limits on their pupils' entry to universities, and redistribute private schools' assets to the state sector.
The shadow chancellor said that he hoped the campaign will "pick up support and eventually become Labour Party policy".
He said: "I think we can gain a large number of votes on the issue, because I believe people think that everyone should have a fair start in life, and that starts by making sure that we all have the same access to education facilities.
"We know that our society is grotesquely unequal, and part of the reason for that is because the inequalities in education particularly in private schools, where large amounts of money are spent on a privileged few."
However, an editorial in the Telegraph argues that a Labour war on private schools would be "dangerous".
They write: "Labour has come a long way since the Blair years, when it believed in expanding opportunity: now it wants to limit it.
"This is just another reason why this radical, neo-Marxist movement must be beaten at the next election."
The Telegraph says that parents should have the right to choose how their children are educated.
The editorial argues: "Private schools are an enormous boon to this country.
"They save the taxpayer's money by taking children out of the state system; they generate billions; they employ thousands; and they promote subjects that are essential to our economic future.
"Across the world, independent schools are regarded as definitively British and part of our international brand. They are another reason why so many skilled people choose to come here and raise a family."
Around six to nine per cent of schoolchildren in the UK are privately educated. According to the Independent Schools Council, the UK independent sector educates around 630,000 children in around 2,500 schools.
Despite making up a small proportion of the population, more than half of Britain's senior judges, top civil servants and diplomats attended private school.
In 15 of the UK's top universities, the proportion of privately-educated students is more than 30 per cent.
Nearly two-thirds of Boris Johnson’s cabinet were privately educated. Ministers are nine times more likely to have attended a fee-paying school than the general population, according to the Guardian.
The average annual fee for a day pupil is £17,200. The fee is lowest for private schools in the North West, averaging around £11,000 per year.