Oxford insists background is not a factor
The universities accept the statistics - but say the problem is that too few state school pupils apply.
They say they are making great efforts to change things.
The issue is not new but became headline news in May after it was reported that Laura Spence, a pupil from a comprehensive school, had been refused a place to study medicine at Magdalen College, Oxford.
Journalists seized on the fact that Laura, from Monkseaton Community High School in Whitley Bay, Tyne and Wear, then won a scholarship to Harvard in the United States.
It subsequently emerged that she had been offered medical places by other prestigious UK universities to which she had also applied, but had turned them down.
But by then her case had been used by the Chancellor, Gordon Brown, as an example of elitism in Oxford working against state schools.
It was, he said, "an absolute scandal", an example of "an old establishment interview system denying her access".
His comments caused outrage in Oxford, which only days beforehand had renewed a campaign to try to attract more gifted state school sixth formers.
Statistical evidence
The university's vice-chancellor, Colin Lucas, said it was "constantly seeking out the most able students to come and study with us - whatever school they have been to, whatever their background, whatever their accent".
The debate widened, fuelled by damning official statistics highlighted in a report by the Sutton Trust, which works to try to get more bright state students into the best universities.
Identifying 13 "top" universities on the basis of newspaper league tables of academic performance, it pointed out that children from independent schools accounted for 39% of the intake.
If the figures were to have reflected the numbers who were best qualified, independent schools should have accounted for only 28%.
And the chances of getting into a top university were 25 times greater for someone from an independent school than from a lower social class or poor area - almost double what it should have been.
Background influences
But the universities' complaint - that not enough bright state school pupils apply - also seems valid.
Oxford pointed to its own admissions statistics: of the 7,852 UK students applying last autumn for places in 2000, 56% came from state schools.
It made 53% of its offers to those from the state sector - still a slight gap, but nothing like so glaring.
Prestigious universities - such as Oxford, Cambridge, Bristol - have been making efforts for some time to persuade more state schools' pupils to apply for places.
There are brochures, summer schools, "outreach" schemes aimed at teachers as well as pupils, mentoring by existing students - the broad message being: "We are not bastions of elitism and privilege, you could fit in here too."
A key issue is that having top grades at A-level is not enough - it is a given. To further differentiate the best of the best, many universities also interview candidates.
But the argument is that this discriminates - favouring those from schools and families with a tradition and expectation of getting into the most prestigious institutions.
Their backgrounds, the argument goes, allow them to come across more confidently. They are also likely to have had coaching in interview technique from relatives, friends - and their schools.
These schools are also more likely to be believed when they predict their pupils' grades because they have a well-established track record.
There have been various suggestions for changing the system: