By Angela Harrison
BBC News education correspondent
The targets are set by the bodies which fund universities
Oxford and Cambridge universities have both increased the proportion of state school pupils they take - but are still falling short of targets.
Official figures show that in 2008-9, 54.7% of Oxford's new undergraduates were from state schools. At Cambridge, 59.3% of new students were.
About 17% of sixth-formers in England are educated in the private sector.
All universities in the UK are set "targets" - known as benchmarks - on admitting state school pupils.
These are set by the bodies which fund universities.
For 2008-9, both Oxford and Cambridge had a target for just under 70% of new full-time undergraduates from state schools.
The way the targets are calculated changed for that year, leading to lower ones for both institutions - but both still say further refinements are needed.
Both have shown an increase in the proportion of state school pupils admitted, data released by the Higher Education Statistics Agency (Hesa) show.
In 2007-8, Cambridge admitted 57% of its new undergraduates from state schools and Oxford 53.4%.
A spokeswoman for Cambridge University said: "The university welcomes the way in which this year's performance indicators reflect its efforts on widening participation, with state sector admissions reaching a high point of 59.3%, up more than 2 percentage points on the previous year's figure.
"The university makes great efforts to ensure that bright students from all backgrounds are admitted to Cambridge, and we are committed to maintaining those efforts in the future."
The benchmarks take into consideration the subjects a university offers and the qualifications needed to apply.
A spokeswoman for the University of Oxford said: "'There are many economic and social factors which can prevent students reaching their full academic potential by 18.
"For our part, we are doing our utmost to encourage academic ambition from a young age by working with students from 11 up, and by working closely with parents and teachers."
The data shows that the majority of Russell Group universities did not meet the benchmarks on admitting state-educated pupils.
Cardiff, Glasgow, Liverpool, Queen's University Belfast, Sheffield and Southampton were the only Russell Group universities to exceed their targets.
Dr Wendy Piatt, director-general of the Russell Group, said universities in her group would continue to do everything possible to increase participation from under-represented groups.
She added: "The issues of low aspirations, lack of high-quality advice and guidance and, most importantly, under-achievement at school still remain significant barriers to participation and can only be tackled by agencies and institutions across the board."
In Scotland, St Andrews University took 60.7% of its new entrants from state schools, against a target of nearly 74%.
Edinburgh University had 70.8% of its new intake from the state sector, against its target of 78.7%.
Both showed a slight improvement on the year before.
In Wales and Northern Ireland, all universities exceeded their targets.
Other universities in England highlighted by Hesa as being down against their targets on state school admissions included Bristol (60% against a target of 74.9%), Durham (59.2% against a target of 74.6%), Exeter (70.9% against a target of 78.7%) and Oxford Brookes University (71.6% against a target of 90%).
Hesa data released on Thursday also includes drop-out rates.
Across the UK as a whole, 7.2% of young university entrants had dropped out of higher education within a year of starting, the Hesa figures for 2008-9 show.
In England, the drop-out rate was 6.9%, in Wales it was 7.4%, Scotland 8.3% and Northern Ireland 9.9%.
At Oxford, the rate was 1% and at Cambridge it was 0.8%.